. s : = tinted ae int Fe oan es
= eee . Romero ee
2 = athe Nene mgr —
eS +
Sain Sn Sie, Pm ee SM
fap es
ea tne ih a
asie
Wile
ON,
— Mifcellanea Curtofa :
¢
CONTAINING A
COLLECTION
Of fome of the Principal
PHENOMENA
NATURE
Accounted for by the Greateft Philo-
fophers of this Age ;
BEING THE
Moft Valuable Difcourfes, Read and Deliver’d to
the Royal Society, for the Advancement of Phyfi-
jel and Mathematical Knowledge.
VOL. I.
To which . added,
The LAWS of Stereographick Projection,
Laid down and Demonitrated, by Fa.
Bp Piedgfor, F.R. 5.
LONDON,
Printed by F. AZ for R. Smith, at the Bible, under the
— ~Piarza of the Royal- “Exchange i in Cornhill. 1708.
; ' ,
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AN an « toe
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Pr # MRL ee Feory r
De Oe Be Mb ha SERGE a Gls Sei
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ertien in
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nee
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haya dyin asad thro miei sats pecan ne rman
uoxe
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: ‘ ‘ Hiei U : ne ah ain
K , i $A sill Una nti yok rank veer DAHURP rR PTI nt
oa } i h an ; a a
‘ ye . ' Ri
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i . be
\ Cae) alg 1
Me a ry
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‘ f ,
R x ¢ \
Ms u ’ i
Calculation of the Credibility of Hu-
fh mane Teftimony. | Page 1
A Letter from the Reverend Dr. Wallis, Pro-
- felfor of Geometry in the Univerfity of Oxford,
and Fellow of the Royal Society, London, to
_ Mr. Richard Norris, concerning the Colle-
_ Gion of Secants ; and the true Divifion of the
Meridians in the Sea-Chart. Ms, 9
‘An eafie Demonftration of the Analogy of the
_ Logarithmick, Tangents to the Meridian Line
or fum of the Secants 5 with various Methods
"for computing the fame to the utmoft Exattnefs,
byE. Halley. _ “20
‘A moft compendious and facile Method for Con-
ftrutting the Logarithms, exemplified and de@
monftrated from the Nature of Numbers, withe
out any regard to the Hyperbola, with a {pee-
dy Method for finding the Number from the
Logarithn: given. By KE. Halley. 37
“A Solution given by Mr. John Collins, of 4
~ Chorographical Problem, Propofed by Richard |
-. Townley, E/7; oa uel
The Solutions of three Chorographic Problems, by
4 Member of the Philofophical Society of
Oxford. ne | 8
| t “An Arithmetical Paradox, concerning the Chances
of Lotteries by ah os cies Francis Rob-
Ja 2
: erts
aa
ee
the CON TEN Pa 3
_.etts, £/qs., Fellow of the. R:S.- - ae 65.
AA New, Exatt and Ealie Method, of finding the
_ Roots of any Equations generally, and that
without any previous Reduttion, By Edm.
_ p&lalley.. = : a oe 712
AA Differtation concerning the Comftruttion of So-
‘lid Problems, or Equations of the third or
fourth Power, by the help of one (given) Para-
bola and a Circle. By Edmund Halley. 89
‘A Difcourfe concerning the Number of Roots,
in Solid and Biquadratical Equations, as alfo
of the Limits of them. ByE. Halley. 101
Some Iluftrious Specimens of the Dottrine of
_ Eluxions, or Examples by which w clearly
fheton the Ufe and Excellency of that Method
in folving Geometrical Problems, By Ab.
_ De Moivre. | 128
A Method of Squaring fome Sorts of Curves,
Or Reducing them to more fimple Curves. By
A. De Moivre, 2.5.5. a AO
Tio Problems « VIZ. concerning the Solid of
Leaft Refiftance, and the Curve of Swifteft
Defeent. Salad by J. Craig. 159
The Quadrature of the Logarithmical Curve.
By j. Craig. ys rte mae 2
A Theorem concerning the Prp>-tion of the Time
“that a heavy Body [pends in defcending thro’
a right Line joining two given Points, to the
(fhorteft) Tze, in which it paffes from the
one to the other of thefe Points, by the Arch
Of 4 Cyelord, 8. | 168
An Extrat of a Letter fromthe Reverend Dr.
John Wallis, to Richard Waller, E/g;. Se-
cretary to the Royal Society, concerning the
Spaces in the Cycloid, which are perfectly
| Buadrables : 171
, ee BE lar *
hy meets
me fhe CON PEN'ES.
Fhe Quadrature of a Portion of the Epycloid. By
eaar Cafwell.: é ATF
A General Propofition, fhewing the Dimenfion of
the Areas iv all thofe kinds of Curves which
are defcrib’d by the Equable Revolution of .2
Circle upon any Bafis, either a Rettilineal or
a Circular one. By Edm. Halley. 177
A Method of Raifing an infinite Multinomial to
_any given Power, or Extratting any given Root
of the fame. By Mr. A. De Moivre. 183
A Method of Extratting the Root of an Infinite
Equation. By A. De Moivre, F.R.S. 191.
An Experiment of the Refrattion of the Air, ©
made at the Command of the Royal Society,
March 28. 1699. By J. Lowthorp, 4.124.196
A Difcourfe concerning a Method of Difcovering
the True Moment of the Suns Ingrefs into the
_ Tropical Sines. By BE. Halley. 202
AA Seale ef the Degrees of Heat. 215)
“The Properties of the Catenaria. By David
. Gregory, AZ, D. Savilian Profeffor of Aftro-
nomy, and F.R.S. 219
Of the Quadratures of Geometrically irrational -
Figures, By }. Craig. 251!
Concerning the Apparent Magnitude of the Sun
~ and Moon, or the apparent diftance of two
Stars when nigh the Horizon, and when high-
er elevated. 263
The Sentiments of the Reverend and Learned Dr.
_* John Wallis R. S. Soc. upon the aforefaid
Appearance, communicated ia a Letter to the
Publifher. e275
A Demonftration of an Error committed by com-
mon Surveyors in comparing of Surveys,taken at
long Intervals of Time, arifing from the Vari-
ation of the Magnetick, Needle, by William
Ry ee ee te Moly-
a
7 arom
SN eee TR Sree Oa st SED
- ~ =
The CONTENTS. —
Molyneux. E/q,; F. R. S. oop 2.
‘A Propofal concerning the Parallax of the fixed
- Stars, in Reference tothe Earths Annual Orb.
In feveral Letters of May the 2d. June 29.
and July 20. 1693. from Dr. John Wallis
to William Molyneux Efg,
292
Why Bodies diffolved in Menjftrua Specifically
lighter than themfelves, fwimtherein? 300
Of the weight of a Cubic foot of divers grains,
&c. wyd ina Veffel of well-feafowd Oak,
whofe concave was an exatt cubic foot. By
the direttion of the Philofophical Society at
Oxford. 308
“AA Letter of Dr. Wallis to Dr. Sloane, concern-~
ing the Generation of Hail,and of Thunder and
Lightning, and the Effetts thereof. 315
A Synopfis of the Aftronomy of Comets. I
A Geometrical Differtation Concerning the Rain-
bow iz which (by a dirett Method) is fhewn
how to fiad the Diameter of each Bow, the Pro-
portion of the Refraction being given: Together
~ with the Solution of the Inverfe Problem, or
bow to find the Ratio of the Refrattion, the
Diameter of the Iris being given. By Edm-
Halley, F. R. 5. : "25
+
Tabella Poteftatum.
$ | t uz Ww 3 : y
L 27 ol a’-+ol af e-l-2ail a’ ¢ e-4-351 a* e+-351 aett-oil a%e% 71 acti’
hz&—kaot-6k a’ e-|-15k a* ee ey ee ka e-}. k e&
hzi hash at e-t-1oha? e e-f-10ha? e-- sha eh e?
getagattag a et Opa? cel aga er g
friafa-sfa eq 3fa ee fe
dzt=da’trada el dee
czwmcoa+ ce
\
This refers to Page 80 in this Volume.
ADVERTISE MENT.
". 'There is in the Prefs and will Speedily —
be publifh’d, A Collection of the Travels,
Antiquities and Natural Hiftoriesof Coun- ©
tries, as they have been deliver’d in to
the Royal Society; collected from the:
Philofophical TranfaCtions into one Voe
fume in Oftavo. ‘a
— Mifcellanea Curiofa.
PART IL.
A Calculation of the Credibility of
5 — Humane Tefiimony. _
Oral Certitude "Ab/olute,is that in which
the Mind of Man entirely acquief
ed . ces, requiring no further Affurance ‘
As if one in whom I abfolutely confide, fhall
bring me word of 12060/. accruing to me by
Gift, or 4 Ship’s Arrival, and for which there-
fore I would not give the leaft valuable Con-
fideration to be Enfur’d. |
Moral Gertitude Tacompleat, has its feveral
Degrees to be eftimated by the Proportion it
‘bears to the Ab/folure. As if one in whoml
have that degree of Confidence, as that I would
“not give above One in Six to be enfur’d of the
‘trath of what he fays, fhall inform me, as a-
‘bove, concerning 1200/. I may then reckon —
that Ihave as good as the Abfolute Certainty
of a 1000 /. or five fixths of Abfolute Certainty
for the whole Summ. ,
r 5 Bs nl , The
mes a
ss i
2 — Mifcellanea Curtofa.
The Credibility of any Reporter is to berated —
(1) by bis Jvtegrity or Fidelity; and (2) by his”
Ability: and a double Ability is to be confider-
ed; both that of Apprebending what is deliver’d,
and alfo of Retaining it afterwards,. till it be
tranfinikted. (4) F SAS VR aya:
‘ What follows concerning the Degrees of
© Credibility, is divided into Four Propofitions.
‘The Two Firft, refpe@ the Reporters of the
¢ Narrative; as they either Tranfinit Succe/five-
“ly, or Atteft Concurrently : the Third, the Sub- —
‘ veft of it; as it may confift-of feveral Articles: —
© and the Fourth, joins thofe three Confiderati-
‘ ons together, exemplifying them in Oral and
#
© in Written Tradition:
PR OPS PE:
Concerning, rhe Credibility of a Report, made by
‘Single Succeflive Reporters, who are equally Cre-
dible. ;
_ Let their Reports have, each of them, five.
Sixths of Certainty ; and let the firft Reporter.
give me a Certainty of 1000/. in 1200 /. it is,
plain, that the Second Reporter, who delivers.
that Report, will give me the Certainty but of
éths of that 1000 /. or the 3th of ths of the.
full Certainty of the whole 12002... .And fo.a
Third Reporter,who has it from the fecond,will
tranfinit to me but zths of that Degree of Cer-
tainty, the Second would have deliver’d me,
a) a i s 7 z |
. That is, if, 4, be put for the Share of Affu-_
rance a fingle Reporter gives me; and, c, for,
that which 18 wanting to make shat Alfurance .
gompleat; and 1 therefore fuppos’d to have —
A
Og ae me
ie
: ae
a
Mifeellaiew Curiofa. 2
& : :
oC Certainty from the Firft Reporter ; I
— gt : Aa. .
fhall have’ from ‘thé Second,—=='; from the
os bee a” a+”
Third, gers",
»; And accordingly, if, 2, be == 100, and c—=6)
(the number of Pounds that an 1oo/. put out
-t0 Intereft,. brings in at the Year’s.end;.) and
-comfequently my, Share of Certainty from One
‘Reporter, -be ==i5< 5 -which is the, prefent va-
due of any Summ to be paid a Year hence: The
Proportion of Certainty coming to me froma
Second, will be $2 multiplyed by +3% Gyvhich
is.the, prefent: Value of Mony to be paid after
two-Years;,) and that from.a Third-hand. Re-
‘porter, <= iar «thrice multiplied into it felf ;
‘the Value of Mony payable at the end of
Meee Beats) OC. ks
YS "3° Eorollary.
~ And ‘therefore, as at the Rate of 6 per Cent.
Intereft, the prefent Value of any Summ pay-
able after Twelve Years, is-but half the Summ
So if the Probability or proportion of Certi-
anes, tranfmitted by each Reporter, be ize;
the Proportion of Certainty after Twelve fuch
_Tranfmiffions, will be but as a half; and it will
‘grow by that Time an equal Lay, whether the
peat be true or no. Inthe fame manner, if
‘the Proportion of Certainty be fet at ¥25 it
will come to half from the 7orh Hand: And if
a tea) from the 695th. |
it
gaia Po Be PROP.
43 Mifcellanea Curiofa.
PROP? at
Concerning Concurrent Teftificatzons.
If Two Concurrent Reporters have, each of
them, as ¢ths of Certainty ; they will both give
me an Aflurance of 2iths, or of 35 to one: If
Three ; an Aflurance of 312, or of 215 to
one. | '
For if one of them gives a Certainty for
1200 1. as of 3ths, there remains but an Aflu-
rance of {th, or of 200/. wanting to me, for
the whole. And towards that the Second At-
tefter contributes, according to his Proportion
of Credibility: That is to ths of Certainty be-
forehand, he adds ths of the th which was
wanting: So that there is now wartting but
sth of a &th, that is 3th; and confequently I
have, from them both, 34ths of Certainty: So
from Three, 233, Gc. age 3
That is, if the firft Witnefs gives me —
a--C
of Certainty, and there is wanting of it
Cy
AKC 5
the Second Attefter will add “— of that £5
a-~c a*Kc 5
and confequently leave nothing wanting but
c ’ a c? seeks
= of that i == ee like: man-
oy ac 2
ner the third Attefter adds his 2— of that
.2
e / ce a Stes
» and leaves wanting only —— -@ce. ©
Corol-
Pe.
Mifcelonea Curtofa. *
. } 9195/1. . Corollary. :
Hence it follows, that ifa fingle Witnefs
fhould be only fo far Credible, as to give me
the Half of a full Certainty ; the Second ofthe
fame Credibility, would (joined with the firft)
give me 3ths; a Third, gths, cc. So that the.
Coatteftation ofa Tenth,would give me 3: 4ths
of Certainty ; and the Coatteftation of aT wen-
tieth, to35223 or above Two Millions to one,
PRO P. 1h.
Concerning the Credit of a Reporter for a Particu-
lar Article of that Narrative, for the whole of
which he is Credible in a certain Degree.
Let there. be Six Particulars of a Narrative
~ equally remarkable : If he to whom the Report
vis given, has %ths of Certainty for the whole,
--or Summ, of them; he has 35 to one, againft
the Failure in any One certain Particular.
» For he has Five to One, there will be no
Failure atall. And ifthere be, he has yet a-
nother Five to One, that it falls not upon that
-fingle Particular of the Six. That is, he has
4ths of Certainty for the whole: and of the seh
wanting, he has likewife $rhs, or ,irhs of the
_. whole more ; and thereforethat there will be
no Failure in that fingle Particular, he has
_ gths and $4ths of Certainty, or 45 of it.
Bag
a5
Tn General, if <—-be the Proportion of Cer-
. i -
_ tainty for the whole; and ——— be the chance
> of the reft of the particular Articles m, againit
B 3 fome
te AP ae Ag i piiraiuiaeah aie. nie
ee: MifeeNanea Cui
fome one, or more of them 7; there will be no-
thing wanting toan abfolute Certitude ,againit —
the not failing 10 sp few pe Articles, 2, a
Ce g te Pies er iy re»
jails meme te {/) i Vee
marhaneacthe r ay aT
EON ee eee
PROP. Wwe
Concerning the Truth of es Oral or Written
Tradition, (in Whole, or in Part,) Succeflively-
tranfinitted, and alfa Coattelted by feveral Suc-
ceffions of Lage, |
ae 1) Suppofing. the ‘Tranfinidfidn oh: an a Oral
and Narrative to be fo. performed by. a Succef- -
fon of Single Men, or joined in Companies, as
othatveach: Trantinit a } after the “Narrative
has ‘been! kept ‘for Tweaty. Years, impairs the
Credit of ita /th part ; and-that confequently
‘at the Twelfth: Hand; oratrthesendiiof. 240 -
Years, its certainty is redaced to a: Half; and
there grows then‘an even Lay (by the Corollary
of tke fecond Propofiriar) againft the Truth of the
Relation | Vet if we farther. fuppofe; thatthe
-fame Relation is Coatteltet by Nine other feve
yal Succeflions,tranfinitting alike each of them ;
the Credibility of it when they are all found:to
agree; wall (by the Corollary of the firft Propofition)
be as 4333. or Certainty, or’ above a Thoufand
to one: and if we ‘fuppofe aCoatteftation\ of
| Ninetcen, the Credibility ei it Nie be, as a~
-bove' Two Millions ‘to’ One-- aD af
(2) In Oral Tradition as a “Sinate Mah is
fabject to much Cafuality, fo a Company of
Men cannot be fo eafily fuppos’d to join; and
therefore: the Credibility “_ thsi: aera
ae E «eb Ne ts.
i — Mifcellanea Curiofa. m7
' 43ths,may poflibly be judged too high a Degree
for an Oral Conveyance, to the diftance of
Twenty Yeats. But in Written Tradition, the
‘Ghances againft the Truth or Confervation‘of
a fingle Writing, are far lefs, and feveral'Co-
‘pies may alfo be eafily fippos’d to concur ;and
thofe fince the Invention of Printing exactly
the fame: feveral’ alfo' diftin® ‘Succeffions~ of
“fach Copies may be as: well fuppos’d, taken by
‘different Hands, and preferv’d in different
‘Places or Languages. pre: sik te
‘And therefore it Oral Tradition by anyone
‘Man or Company of Men might be fuppos’d
to be Credible, after Twenty Years at ‘2zhs
of Certainty ; or but °,ths; or $ths: a' Writ-
ten Tradition may be well imagin’d to conti-
nue, by the Joint Copies that may be taken of
it for one Place, (like the feveral Copies of the
fame Impreffion} during the fpace of a 100, if
not 200 Years ; and to be then Credible at
724¢hs of Certainty, or at the Proportion of a
Hundred to one. And then, feeing that the
Succeflive Tranfmiffions of this 33° of Certain-
ty, will not diminifh it to a Half, until it paf-
des the. Sixty. nioth Hand ;..(forit..awill-benear
Seventy Years before the Rebate of Mony, at
that Intereft, will fink it to half: ) It is plain,
that written Tradition, if preferv’d but by a
fingle Succeflion of Copies, will not lofe half of
its full Certainty, until 70 times a Hundred Cif
not two Hundred) Years are paft; that is, Se-
ven Thoufand, if not Fourteen thoufand Years;
_and further, that, if it be likewife preferv’d
by Concurrent Succeffions of. {uch Copies, its
ccpbeing at that Diftance may be even en-
-dreas’d, ahd grow far more certain from the
feveral agreeing Deliveries at the end of Se-
B 4. venty
8 Mifcetlanea Curtofa.
venty Succeffions, than it would be at the very —
firft from either of the Single Hands. 4
(3) Laftly,in ftating the Proportions of Cre-
‘dibility for any Part or Parts of a Copy, it may
_be obfery’d ; that in an Original not very long,
good Odds may be laid by a careful Hand,that ©
the Copy fhall not have fo much as a Literal
Fault: But in one of greater Length,that there
may be greater Odds againft any Material Er-
ror, and fuch as fhall-alter the Senfe; greater
yet, that the Senfe fhal] not be alter’d in any
confiderable Points; and ftill greater, if there
be many of thofe Points, that the Error lights
not upon fuch a fingle Article; as in the Third
Propofition. a Hl hi
| Mifcellanea Curiofa. —g
A Letter from the Rewereni Dr.
_ Wallis, Profeflor of Geometry zx
_ the Univer fity of Oxford,and Fel-
opi the Royal Society, London,
_ to Mr. Richard Norris, concern-
ing the Collection of Secants ; ; and
: the true Divifion of the Mer idians
_ an the Sea-Chart.
N old enquiry, (about the Sum or Aggre-
gate of Secants) having been of late mo-
yed a-new ; I have thought fit to trace it from
_ its Original : with fuch folution as feems pro-
per to it: Beginning firft with the general
Preparation ; and then applying i it to the Parti-
cular Cafe.
J : General Preparation.
ai bi “Beeaufe Curve lines are not fo eafily ma-
-naged as Straight lines: the Ancients, when
they were to confider of Figures terminated
(at leaft on one fide) bya ‘Curve line (Con-
‘vex or Concave) as AFKE; Fig.1.2. Tab.1. did
oft make ufe of fome fach expedient as this
following, (but diverfly varied as occafion re- .
quir’d.) Namely,
2. By Parallel. Straight. lines, as AF, BG,
CH, &c.. (at, equal or. unequal diftances, as
there was occafion,) they parted it into fo ma-
“ny:
ao Mifcetanea Curiofa.
ny Segments as they thought fit; (or fu ippofed
at to-be fo parted)"
3. Thefe Segments were fo many manting one,
as(was the number of thofe Parallels.
"4. To each of thefe Parallels, gta one ;
they fitted Parallelograms, of fuch breadths as
were the Intervals (équal or unequal) between
each of them, (refpectively) and the next fol-
dowing. Which formed an Adfcribed Figure
madet up of thofe Parallelograms. 1)
\..§- And,.if they, began with \the Greateft
Gid therefore negle@ed.the leaft) fuch Figure
was. Circumfcribed, (as Rig) rt.) and therefore
Bigger than the Curvilinear propofed..
. If with the Leaft (negle@ting the great-
ay . the Figure was Infcribed (as Fig, a.)
vant therefore hefs than that propofed-
vg. Butpasethe amber of Segments wit in-
weed {ets (aiid thereby ‘their breadths' diminifh-
ag ): the difference of the Cireumferibed from
ithe Infcribed (and therefore of either ‘from
that propofed) did continually decreafe, fo as
at laft to be lefS than any affigned.-1)2 15.89
8. On which oy, Evene! their Method
of Exhauftions.:
9. In cafes heen the Breadth of the Pa-
rallelograits,. or Intervals of the'Parallels, is
' mot to be confidered, but their Iength only ;
(or, which is‘much: the fame; ‘wherethe Inter-
yals/are'all the fame, and eachreputed ="1.)
‘Archimedes. Cinftead oF Inferibed®a nd -Circum-
feribéed SFigurés) ufed to fays AU excepe “the
Greatef,-and’ AWexcept ‘the sing 9 ic Cae gai iT.
Lin. cae
Particular: Cafe
dk «3 4 thle 48 be well’ known} edist? ets
Terreftrial’ ¢) all ‘the Me dans’ theet ae
MaifeelaneaCuriofa. = BX
‘the Poles; (av EP. EP, Fig 3.) whereby the
Parallels to the Equator , as they be nearer to
the Pole, do continually ‘decreate.
1p. And hereby a degree of Longitude in
-fach Parallels, is lefs than a degree of Longi-
tude in the: Equator, of a degree’ “of Latitude.
12. And) that,:in. fuch proportion, as is the
-€o-fine of Latitude (which is the femidiamiter
-of fuch Parallel,) to the Radius of the Globe,
or of the Equator.
13./Yetchath it been thought fit (for fome
reafons) to reprefent thefe Meridians, in the
» Sea Gheire, by Parall eb ftraight lines 51as EP,
ed 6
TH, AMWhereby, es oe allel to the Poehter
(as L A) was reprefented in the Sea-Chart,
(as la,) as equal to the Equator £E: and.a
degree of Longitude thereia, as lar ge as in the
Equator.
1geBy thisimeans, each dears: of Lanet-
- tude in) fuch Parallels, was’ increafed, beyond
+
its jaft proportion, at fuch rate as “the Equa-
tor (or its Radius) is greater than fuch Paral-
lel, Cor the Radius thereof.)
16. But, in the Old Sea- Charts, the déevers
of: Latitude were yet reprefented ‘(as they are
in’ themfelves) ‘equal! to each other ; ‘and, to
thofe of the Equator.
17. Hereby, amongft many other Inconve-
-miencres, (as Mr. Edward Wright obferves, in.
\ his: Correttion.of Errors in Navigation, firft pu-
blifhed in the Year 1599,) the reprefentation
“of Places remote froin the Equator, was fo di-
' ftorted, in thofe Charts, as that (for inftance)
an Tland i in the Latitude of 6o degrees, (where
the Radius of the Parailel is but half fo great
ee f oe of the Equator) would have its Length
(frem
‘42 «=Mifcellanea Curtofa, —
(fromEaft to Weft)in comparifon of its Breadth
(from North to South) reprefented in a dou-
ble proportion of what indeed it is. ,
18. For rectifying this in fome meafure (and
-of fome other inconveniences)Mr.Wright advi-
feth ; that (the Meridians remaining Parallel,
-as before) the degrees of Latitude,remote from
the Equator, fhould at each Parallel, be pro-
ah in like proportion with thofe of Longi-
tude. :
19. That is; As the Co-Sine of Latitude,
(which is the Semi-diameter of the Parallel)
to the Radius of the Globe, (which is that of
the Equator: ) fo fhould be a degree of Lati-
“tude, (which is every where equal to a degree
of Longitude in the Equator,) to fuch a de-
gree of Latitude fo protracted (at fuch diftance
‘from the Equator;) and fo to be reprefented
in the Chart. |
20. That is, every where, in fuch proporti- -
on asis the refpective Secant (for fuch Lati- —
tude) to the Radius. For, as the Co-fine, to |
the Radius; fo is the Radius to the Secant (of
the fame Arch or Angle ;) as Fig. 4.2-R::
R
- 21. Sothat ( by this means) the pofition of
each Parallel in the Chart, fhould be at fuch ©
diftance from the Equator, compared with fo
many Equinottial Degrees or Minutes, (as are
thofe of Latitude,) as are all the Secants (ta-
ken at equal diftancesin the Arch) to fo many
times the Radius. nn
22. Which is equivalent, (as Mr. Wright
there notes) to the Projection of the Spherical
furface (fuppofing the Ey-at theCenter)on the |
concave furface of a Cylinder, erected at right
' Angles to the Plain of the Equator.
Hayek) : 24. And
—Mifcellanea Curiofe: 12
_ 23. And the divifion of Meridians, repre
fented by the furface of a Cylinder ere&ted (on
the Arch of Latitude) at right Angles, to the
Plain of the Meridian (or a portion thereof.)
The Altitude of fuch Projection, (or portion of
- fuch Cylindrick furface) being (at each point
of fuch Circular bafe) equal to the fecant (of
Latitude) anfwering to fuch point. As Fig. §.
. 24. This Projection (or portion of the Cy-
lindrick furface) if expanded into a Plain, will
be the fame with a Plain Figure, whofe bafe is
equal toa Quadrantal Arch extended (or a
‘portion thereof) on which (as ordinates) are
erected Perpendiculars equal to the Secants,
anfwering to the refpedtive points of the Arch
fo- extended: The leaft of which (anfwer-
ing to the Equinoétval) is equal to the Radius;
and the reft continually increafing, till (at the
Pole) it be infinite. As at Fig. 6.
25. So that, as ER/L. (a Figure of Secants
erected at right Angles on EL, the Arch of
- Latitude extended,) to ERRL, (a rectangle
onthe fame bafe, who’s altitude ER is equal
to the Radius; ) fo is E Z (an Archof the E-
‘quator equal to that of Latitude,) to the di-
fines of fuch Parallel, (in the Chart) from the
Equator.
26. For finding this diftance, anfwering to
each degree and Minute of Latitude; Mr.
Wright (as the moft obvious way) adds all the
Secants (as they are found calculated in the >
Trigonometrical Canon) from the beginning,
! rae degree or Minute. of Latitude propo-
fed. | ;
- 29. The fum of all which, except the Great-
eft, (anfwering to the Figure Infcribed) is too
Little: ‘The fum of all except the Leaft, (an-
ee ; | fwering
14 Mifcellanea Curwfa-
fwering to the Circumfcribed,) is too Great,
(which i is that he follows : ) And it would be
nearer to the Truth than either, if (omitting |
all thefe) we.take the intermediates ;-for Min.
ly fs 2% 335 @c. or (the doubles " of thefe)
Min. 1,3, 5,7, ce. Which yet (becaufe on
the Convex fide of the Curve) would be fome-
what too Little. al
28. But any of thefe ways are exact enough
for the ufe intended, -as. creating No fenfible
difference in the Chart. |
29. If we would be more exad ; Mr. oie
tred direéts (and fo had Mr. Wright done be-
fore him) to divide the Arch into parts yet
fmaller than Minutes, and. calculate Secaiits
fuiting thereunto. ”
30. Since the Arithmetick of infinite’ intto-
- Pik b>
without st come toa 7 dorechdnees propor-
tion ;.) Methods have been found for {quaring
fome fuch Figures ; and (particular lyCthe Ex-
terior, Hyperbola (ia a way of continual ap-
proach ) by the help, of an Infinite feries. . As,
in the Philofophical Tranfaclions, Numb. 38, (for
the Month of -Aaguff, 1668, ) And my. Book,
De sie Cap. 5. Prop. 3r. |
. In Imitation whereof, it hath been defi-
at a find) by fome, t that a like Quadrature
for this Figure of Secants (by an Infinite feries
_ fitted thereunto) might be found.
32. In order to which, put we for the Radi-
us of a Circle, R ; the right Sine of an Arch
or, Angle, S,, the Verfed Sine 5, V, the-Co-Sine
(or Sine of the. Complement) ee pEaie eV SW:
apa the Secant, i the Tangent, ‘T. Fig:
ing
. 33. Then
Mifcelianea Curiofa. vf
33. ‘Then i IS, . Se R: ER. W 9 That is, a R%
pag ; the Secant, r ayitoed ? : oy
eee ‘And 2S RT. That 52)SR TS
the Tangent.
prt Now, if we. fuppofe the Radius C P, Fe
. divided into equal Parts, (and each of them, :
ibs R;) and, on thefe, to be erected the Co-,
Sines of Latitude LA:
36. Then are the Sines of Latitude in Arithe,
metick Progreffion.
37° And the Secants anfwering thereunto,.
| Lf=4 =" we
38. But thefe Secants, (anfwering to right
Sines in Arithmetical Progreflion) are not thofe,
that ftand at equal diltance on the Quadran-
tal Arch extended, Hig. 6. |
39. But ftanding at unequal diftances (on
the fame extended Arch; ) Namely, on thofe
points thereof, whofe right Sines (whilft it was:
a Curve) are in Arithmetical Progreilion. As,
Fig. 8.
40. ap! find therefore the magnitude of RE:
Lf, Fig.6. Which is'the fame.with that’ of,
Fig. 8. phate EL of the fame length in
both; however the number of Secants therein
may. be unequal: ) we are. to confider the Se-.
cants, tho’ at unequal diftances: Fig. 8. to be:
the fame with thofe at equal diftances in: Fig:
7. anfwering to Sines in Arithmetical Progref-
fion.
4l. Now thefe Intervals, (or portions of the
bafe) in Fig. 8. are the fame with the inter-
iG cepted Arches (or ‘portions of the Arch) in Fig.
_ -74'For this bafe is but tbat Arch extended. .
A | 42, And
5 z
Saar a aS
16 Mifcellanea Curiofa.
42. And thefe Arches (in parts infinitely
fmall) are to be reputed equivalent tothe por-
tions of their refpective Tangents intercepted
Between the fame ordinates. As in Fig.7- 9.
43. That is, equivalent to the portions of
the Tangents of Latitude. |
44. And thefe portions of Tangents are, to
the Equal intervals in the bafe, as the Tan-
gent (of Latitude) to its Sine.
45. To find therefore the true Magnitude of
the Parallelograms (or fegments of the Figure; )
we muft either protract the equal fegments o
the bafe, Fig. 7. (in fuch proportion as is the
refpective Tangent to the Sine) to make theni
equal to thofe of Fig. 8. :
46. Or elfe (which is equivalent). retaining
the equal intervals of Fig. 7. protract the Se-
cants in the fame proportion. (For, eitherway
the Intercepted Rectangles or Parallelograms
will be equally encreafed) As L 1 Fig. 9.
47. Namely; As the Sine (of Latiude) to
its Tangent ; fo is the Secant to a Fourth;
which is to ftand (on the Radius equally divi-
| ded) inftead of that Secant.
SR “Re R3 Spe ee
S.S(::=-R):: | s7eReso=L M, Fig.o.
48. Which therefore are as the Ordinates |
. kn Qwhat I call Arith. Infin. Prop. 104) Recipro-
ca Secundinorvum: fuppofing =? to be fquares
in the order of Secundajies.
— Mifecilanen Cnriofa. « 19
2 3 5) ' 9 t : S» S$ 1
Boe fits) RGR igstaeeth
; 4] 4 a oe : ~ c R3-S2R. P
49. This becauie oi aera
| hy pata Japa S4.
= *=RY-S',& the Sines +-S°R7R
$, in Arithmetical Pro-
greffion) is reduced (by) 4.54
divifion) into this Iof- R a
nite Series. | it ys 124 Wei!
hy. ae aS
R-- een Re &c ; :
Yo. ‘That is(putting R=1.) hp;
. Ll S*.\- S44. $°, &c. : , )
_ §1. Then.(according to.the Arithmetick of
Infinites) we are to interpret S, fucceflively, by
1 S, 2S, 3 S, &c. till we come to 5, the great-
eft, Which therefore reprefents. the number
of All. |
52. And becaufe the firft Member.doth re-
prefent a Series of Equals; the fecond of Se-
Beats the third, of Quartans, @c. There-
fore the firft Member is to be multiplied by S$;
the fecond, by}S ; the third, by; 5; the
Mourth, by 75; cc.
~ 53. Which makes the Aggregate,
— $tistt+is7+is’+,s’, &=ECLM,
. becaufe S is always lefsthan R= |
may be fo far continued, till fome power of
tording to Mr. Wright’s defign: ) Having the
y
18 Mifcellanea Curiofa.
propofed ParaJlel (of Latitude) given; we are’
to find (by the Trigonometrical Canon) the
Sine of fuch Latitude, and take, equal to it,
CL=s8. And, by this, find the magnitude of ~
ECLM, Fig. 9; thatis, of RELA Fig. 8. —
that is, of REL, Fig.6. And then, as R
RLE (or fo many times the Radius,) toR BE
Lf (the Aggregate of all the Secants;) fo
muft bea like Arch of the Equator (equal to
the Latitude propofed,) to the diftance of
fach Parallel, (reprefenting the Latitude in the
Chart) from the Equator. Which is the thing
required. | |
56. The fame may be obtained, in like man-
ner, by taking the Verfed Sines in Arithmeti-
cal Progreflion. For if the right Sines (as here)
beginning at the Equator, be in. -Arithmetical
Progreffion, as 1, 2, 3, @c. Then will the Ver-
fed. Sines, beginning at the Pole, (as being
their complements to the Radius) be fo alfo. ©
The Collettion of Tanvents.
57. The fame may be applied in like man-
ner, (though that be not the prefent bufinefs,)
to the Aggregate of Tangents, (anfwering to
the Arch divided into equal parts.) = ;
58. For, thofe anfwering to the Radius fo
divided, are S~3 ( taking S. inArithmetical Pro-
greffion.) Steyn SE ae
_ §9- And then, inlarging the, Bafe, (Cas in
Fig. 8.) or the Tangent (as in Fig.’9.) in the
3
proportion of the Tangent to the Sine.
Plate J pageso
: OS ae Bao
Maifcellanea Curiofa. 19
= Ne al SR2: SR2
Ss. = ue ee OE oy Se R2--S2°
60. e. have (by Divifion) this Serles,
: : ins
| R? S)SR? (S,+ So +gor
R4
Sigite S54 P78 Ba a ee
pom
61. That is(putting ee ue 4538
Si. S3 Ss 97 1S, &ce
62. Which (multiplying 3s
thie refpective members by i oy e)
| 3S, 495 = 9 ay: 1.95 &c.) be- he "Ra
-comes |
idle Se : 445% 4.651% &e, 4, $7
Which is the onaalies of Tangents to the
Arch, whofe right Sine is S.
.. 63. And this method may be a pattern for |
the like procefs in other cafes of like nature.
3
An
20 Mifcellanea Curtofa.
An eafte Demonflration of the Analo-—
wy of the Lovarithmck Tangents
to the Meridian Line or [um of tbe
Secants ; with various Methods
for computing the [ame to the utmoft
Exacine|s,*oy E. Halley. |
T is now near 100 Years fince our Worthy
Countryman, Mr. Edward Wright, publith-
ed his Correéttion of Errors in Navigation, a Book
well deferving the perufal of all fuch as defign
to ufe the Sea. Therein he confiders the Courfe
ofa Ship on the Globe, ftearing obliquely to
the Meridian ; and having fhewn, that the De-
parture from the Meridian, is in all cafes lefs
than the Difference of Longetude, in the rats of
Radius to the -fecant of the Latitude, he con-
cludes, That the fum of the Secants of each
point of the Cuadrant being added fucceflively
would exhibit a line divided into Spaces, fuch
as the intervals of the parallels of Latitude
ought to be ina trueSea-Chart,whereon theMe-
-ridians are made parallel Lines, and the Rhombs
or Oblique Courfes reprefented by right Lines.
This is commonly known by the name of the
Meridian Line,which tho’ it generally be called
_Mercator’s, was yet undoubtedly Mr. Wright’s
Invention, (as he has made it appear in his
' Preface.) And the Table thereofis to be met
with in moftBooks treating of Navigation,com-
puted with fufficient exattnefs for the dae
eA | C
) ViifceHanea Curiofa. 21
' * Te-was firft difcovered by Chance, and as far
as licamdearn, firft publifYd by Mr. Henry
Bond, as an addition to Norwood’s Epitome of
Naviration, about 50 Years fince, that the A7-
ridian Line was analogous to a Scale of Lozarith-
mick, Tangents of half the Complements of the
Latitudes. The difficulty to prove the truth
of this Propofition, feemed fuch to Mr. Aer-
_cator, the Author of Logarithmotechnia, that he
propofed to wager a good fum of Mony,againit
whofo would fairly undertake it, that he fhould
not demonftrate either, that it was true of
falfe: And about that time Mr. fobx Collins,
holding a Correfpondence with all the Eminent
‘Mathematicians of the Age, did excite them
tothis enquiry = =
* The firft that demonftrated the faid Azalogy,
was the excellent Mr. Fames Gregory in bis Ex-
ercitationes Geometrice, publifhed Anno 16658.
which he did, not without a long train of Con-
fequences and. Complication of Proportions,
whereby the evidence of the Demonftration is
in a great meafure loft, and the Reader wea-
ried before he attain it. Nor with lefs work
and apparatus hath the celebrated Dr. Barrow,
in his Geometrical Leétures (Lect. XI. App. 1.)
proved, that the Swm of all the Secants of
any arch is analogous tothe Logarithm of the
ratio of Radius -\- Sine to Rad.—Sine, or, which
is all one, that the Meridional parts anfwering
to any degree of Latitude, are as the Logarithms
of the rationes of the Verfed Sines of the diftan-
ces from both the Poles. Since which the in-
comparable Dr. Wallis (on occafion of a Para-
logifm committed by one Mr. Zvorris in this
matter) has more fully and clearly handled this
Argument, as may be feen in Num. 176. of
Ry tid 3 the
22 Maifcellanea Curiofa. q
the Tranfattions. But’ neither Dr. Wallisnor
Dr. Barrow, in their faid Treatifes, have any
where touched upon the aforefaid relation of
the Meridian-line to the Logarithmick Tangent ;
nor hath any one, that 1 know of, yet difcover-
ed the Rule for computing independently the
interval of the Aeridional parts anfwering, to’
any two given Latitudes. a
Wherefore having attained, as I conceive, a
very facile and natural demonftration of the
faid Analogy, and having found out the Rule
for exhibiting the gifference of Meredional parts,
between any two parallels of Latitude, without
finding both the Numbers whereof they are the
difference: I hope I may be entituled to a
fhare in the Improvements of this ufeful part
of Geometry. And firft, let us demonftrate
the following Propofrtion. 6 otenart bor
The Meridian Line is a Scale of Logarithmick,
Tangents of the half Complements of the Latitudes. .
For this Demonftration, it is requifite to
premife thefe four Lemmata.
Lemma. \. In the Stereographick, Projettion of
the Sphere upon the plain of the Equinodiual,
the diftances from the Center, which in this —
cafe is the Pole, are laid down by the Tan-
gents of half thofe diftances, that is, of half
the Complements of the Latitudes. This is
evident from Eucl. 3. 20. |
Lem. Vi. In the Stereographick Projection, the
Angles under which the Circles interfect each
- other, are in all cafes equal to the Spherical.
Angles they reprefent: Which is perhaps as
valuable a property of this Projeétion, as that
of all the Circles of the Sphere thereon appear-.
| ee
Maifcellanea Curtofa. 22
“ing Gircles: But this not being vulgarly known,
-muft not be aflumed without a Demonjftration.
Let EBPL in Fig. 1. Tab. 2. be any great
circle of the Sphere, Ethe Eye placed in Its
Circumference, C its Center, P any point
thereof, and let FC O be fuppofed a plain
erected at right Angles to the Circle E BP L,
on which FCO we defign the Sphere to be
projected. Draw E P crofling the Plain F
CO inp, and p fhall be the Point P projected.
To the point P draw the Tangent A P G
and on any point thereof, as A, erect a per-
pendicular AD, at right angles to the plane
EBPL, and draw the lines PD, AC, DC:
and the AngleAPD fhall be equal to the Sphe-
_rical Angle contained between the plains 4 P
C, DPC. Draw alfo AE, DE, interfe&ing
the plain FC9 in the points 2 and d; and
joyn ad, pd: I fay the Triangle 2d is fimu-
‘Jar to the triangle 4DP. And the Angle apd
equal to the Augle APD. Draw PL, AK, pa-
rallel to FO, and by reafon of the parallels, 4
pwill be toad as AKto AD: But (by Excl.
3. 32.) in the triangle AKP, the angle AKP=
LPE is alfo equal to APK= EPG, wherefore
the fides AK, AP, are equal, and ’twill be as
aptoad fo AP to AD. Whence the angles
DAP, dap being right, the angle APD will
be equal to the angle zpd, that is, the Sphe-
rical Angle is equal to that on the Projection,
and that in all Cafes. Which was to be proved.
This Lemma | lately received from Mr. 46.
de Moivre, though! fince underftand from Dr.
Hook, that he long ago produced the fame
thing before the Society. However the demon-
{tration and the reft of the Difcourfe, is my
own.
C4 Lemma
24 Maifcellanea Curiofa. i!
Lemma Il. On the Globe, the Rumb Lines
make equal angles with every Meridian, and —
by the aforegoing Lemma, they muft like-_
wife make equal angles with the Meridians in
the Stereographick Projection on the plain of the -
Equator; They are therefore,in that Projection,
Proportional Spirals about the Pole Point. .
Lem JV. in theProportional Spiral (Fig.2.)it is
a known property,that the angles BPC, or the.
arches BD, are E.cponent's of the rationes of BP
to PC: for if the arch BD be divided into in-
numerable equal parts, right lines drawn from
them to the Center P, fhall divide. the Curve B
ccC, into an infinity of proportionals ; and all
the lines Pe fhall be an infinity of proportionals
between PB and PC, whofe number 1s equal to
all the points d,d, in the arch BD: Whence
and by what I have deliver’d in the next enfu-
ing Difcourfe it follows, that as BD to Bd, or
as the angle BPC to the angle BPc, fois the
‘Logarithm of the ratio of PB to PC, to the Lo-
garithm of the ratio of PBto Pc.
. From thefe Levmata our Propofition is very
clearly demonftrated :. For by’the firft, PB,
Pc, PC are the Tangents of half the Comple-
ments of the Latitudes in the Stereographick,
Projeétion: and by the laft of them, the diffe-
rences of Longitude, or angles at the Pole be-
tween them, are Logarithms of the ratzones of
thofe Taugents one to the other. “But the Nau-
tical Meridian Line, is no other than a Table
of the Longitudes, anfwering to each minute
of Latitude, on the Rhumb-line, making an
angle of 45 degrees with the Meridian Where-_
fore the Meridian Line is no other than a Scale
of Logarithmick Tangents of the half aeats
) ments
%
Mifcellanea Curiofa. aaa.
_ ments ofthe Latitudes. Quod erat demonftran-
dum.
Coroll. 1.Becaufe that in every point of any
Rhum Line, the difference of Latitude is to the
Departure, as the Radius to the Tangent of the
angle that Rhumb makes makes with the Me-
ridian; and thofe equal Departures are every
where to the differences of Longitude, as the
Radius to the Secant of the Latitude ; it fol-
ows, that the differences of Longitude are, on
_any Rhumb, Logarithms of the fame Tangents,
but of a differing Species, being proportioned
to one another as are the Tangents of the ai-
gles made with the Meridian.
Coroll, 2. Henceany Scale of Logarithm Tan-
gents, (as thofe of the Vulgar Tables made
after Bricgs’s form; or thofe made to Nzapier’s,
or any other form whatfoever) is a Table of
the differences of Longitude, to the fevera] La-
titudes, upon fome determinate Rhumb or o-
ther: And therefore, as the Tangent of the
angle of fuch Rbhumb, to pe Tangent of any
other Rhumb: So the difference of the Loga-
rithms of any two Tangents, to the difference °
of Longitude, on the propofed Rhumb, inter-
cepted between the two Latitudes, of whofe -
half Complements you took. the Logarithm
Tangents. .
And fince we have a very compleat Table
of Logarithm Tangents of Briggs’s form, publish-
ed by Vlacq, Anno 1633, in his Canon Magnus
Triangulorum Logarithmicus, computed to ten
Decimal places of the Logarithm, and to every
ten Seconds of the Quadrant (which feems to
be more than fufficient for the niceft Calcula-
tor) I thought fic to enquire the Oblique angle,
with which that Rhumb Line crofles the or
ridian,
26 Maifcelanea Curiofa.
ridian, whereon the faid Canon of Valcg: pre-
cifely anfwers to the differences of Longitude,
putting Unity for one minute thereof, as in
the Common Meridian Line. Now;-the mo-
mentary augment or fluxion of the Tangent Line
at 45 degrees, is exactly double to the fluxion
of the arch of the Circle, (as may eafily be
proved) and the Tangent of 45 being equal to
Radius, the fluxion alfo of the Logarithm Tan-
gent will be double to that of the arch, if the
Logarithm be of Napier’s form : but for Briggs’s
form, it will be as the fame doubled arch, mul-
tiplied into 0, 43429, &c. or divided by 2,
30258, &c. Yet this muft be underftood only
of the addition ofan indivifible arch,for it cea-
fes to be true, if the arth have any determi-
nate magnitude. ye | |
Hence it appears, that if one minute be fup-
pofed Unity, the length of the arch ofone mi-
nute being ,00029088820866 5721596154, KC.
in parts of the Radius, the proportion will be
as’ Unity to 2,908882, &c. fo Radius to the
Tangent of 71° 1/ 42'? whofe Logarithm is 10.
4637261172071832§204, &c. and under that
angle is the Meridian interfected by that
Rhumb Line,on which the differences of Napier’s
Logarithm Tangents of the half Complements
of the Latitudes are the true differences of Lon-
citude, eftimated in minutes and parts, taking
the firft Four Figures for Integers. But for-
Vlacq’s Tables, we muft fay. . |
As .2302585, &c. to 2908882, &c. So Ra-
dius to 1,26331143874244569212, &c. which
is the Tangent of 51° 38 9”, and its Logarithm
10,101 §10428507720941162, &c. wherefore in
the Rhumb Line, which makes an angle of 51°
33’ 9” with the Meridian, Vlacq’s Logarithm
Tan-
= Ge
Mifcellanea Currofa. ey:
Tangents are the true differences of Lon-
gitude. And this compared with our fecond
Corollary may fuffice for the ufe of the Tables
already computed. :
_ But if a Table of Logarithm Tangents be
made by extraction of the root of the Infiniteth
power, whofe Index is the length of the arch
- you put for Unity, (as for minutes the ,ooo
—-2908882th,&c. power) which we will call a;
fuch a Scale of Tangents fhall be the true Me-
ridian Line, or fum of all the Secants taken
infinitely many. Here the Reader is defired
to have recourfe to my little Treatife of
Logarithms, in the enfuing Difcourfe that I
may not need to repeat it. By what is there
delivered,it will follow, that putting ¢ for the
excefs or defect of any Tangent above or under |
the Radius or Tangent of 453; the Logarithm
of the ratio of. Radius to fiuch Tangent will be
é into ¢—tee |. $ete— freee + +t5,&c.
when the arch is greater than 45 5" OF
Pe intoe + 4ee-+ 323 40+ + 405, &e.
when it is lefS than 45¢r. And by the fame do-
rine putting T for the Tangent of any arch,
and ¢ for the difference thereof from the Tan-
gent ofanother arch, the Logarithm of their
yatio will be
hints ie ne eee Stn
minto rb orp 575 Vie
when T is the greater Term, or
i if tt 4a Peta, Be $
into > — ary} ant as Gr, &e
hen T is the lefler Term: - LLasies
he And if.m be fuppofed 0002908882, &e. =
the
23 Maifcellanea Curwfa.
: r :
a, its reciprocal > will be, 3437574977973493
92326, &c. which multiplied into the afore+
faid Series, fhall give precifely the difference of
Meridional parts, between the two Latitudes,
to whofe half complements the aflimed Tan-
gents belong. Nor is it material from whe-
ther Pole you eftimate the Complements,whe-
ther the clevated or deprefled , the Tangents
being to one another in the fame ratio as their
Complements, but inverted. rate
In the fame Difcourfe I alfo fhewed, that
the Series might be made to converge twice as
fwift, allthe even powers being omitted sand
putting 7 for the fum of the two Tangents, the
fame Logarithm would be
$y st
je -|- ts
7 Ree suas ti a gag
"A Or 1 into a 373 erry 979 te
oe
but the ratio of 7 tot, or of the fum of two
Tangents to their difference, is the fame as
that of the fime of the fum of the arches, to the
fine of their difference. Wherefore, if S be put
for the fie Complement of the Middle Lati-
tude, and s for the fine of half the difference
of Latitudes, the fame Serves willbe
ar. is sige a ee :
iad
wherein, as the differencts of Latitude are
fmaller, fewer fteps will fuffice. And if the
Equator be put for the middle Latitude, and
confequently S==R, and s to the five of the La-
titude, the Meridional parts reckoned from
the Equator will be | Ot as Vibha
§
&
_
Mifcellanea Curiofa. 29
x
a :
—— &c.
sf | S iH sf
Te 3 Pe ie | 3 :
a 317 & 57 @ Sd ar
which is coincident with Dr. Wallzs’s folution
in Numb. 176. of the Philofophical Tranfattions.
And this fame Series being half the Logarithm
of the ratio of R-+-s to R—-s, that is, of the Ver-
_fed-fines of the diftances from both Poles, does
agree with what Dr. Barrow had fhewn in his
Xith. Ledture. a :
“The fame ratio of > tot may be expreffed alfo
by that of the Swm of the Co-fines of the two Lati-
tudes, to the fine of their difference: As likewife
by that of the Sine of the Sum of the two Lati-
tudes, to the difference of their Co-fives: Or by
that of the Verfed-fine of the Sum of the Co-lati-
tudes, tothe difference of the fines of the Latitudes:
Or as the fame difference of the fines of the Latte
tudes, to the Verfed-fine of the difference of the La-
titudes . all which are in the fame ratio of the
Go-fine of the middle Latitude, to the Sine of
half the difference of the Latitudes. As it
were eafie to demonftrate, if the Reader were
not fuppofed capable to do it himfelf, upon a
_ bare infpection of a Scheme duly reprefenting
thefe Lines: 3 Le
This variety of Expreflion of the fame ra-
tio {thought not fit to be omitted, becaufe by
help of the rationality of the Sine of 3097.
in all cafes where the Sum or difference of the
Latitudes is 3097. 6ogr. gogr. 120¢r. or 1§0
degrees, fome one of them will exhibit a fimple
feries, wherein great part of the Labour will be
faved : And-befides 1 am willing to give the
‘Reader his choice which of thefe equippolent
methods to make ufe of ;-but for his exercife
of | fhall
20 M t{[cellanea Curiofa.
fhall leave the profecution of them, and the
compendia arifing therefrom,to his own Induftry.
Contenting my felf to confider only theformer,
which for all ufes feems the moft convenient,
whether we defign to make the whole Meri-
dian Line, or any part thereof, viz. :
¥ S us $
oY BA § s 5 pest § S
— into =-+ —=- —— +
a
EY ar iter Ae rggs eG
Wherein @ is the length of any Arch which
you defign fhall be the Integer or Unity in
your Meridional Parts,(whether it bea Minute,
League, or Degree, or any other,) S the Co-
fine of the Middle Latitude, and s the Sine of -
half the difference of Latitudes; but the Se-
cants being the Reciprocals of the Co-fines,
= will be equal to putting {for the Secant
of the Middle Latitude; and — into = will
caer Ae multiplied by 355 that is by
ffss
3777s
will give the fecond ftep: and that a-
gain by A the third ftep ; and fo forward,
till you have compleated as many places as you
defire. But the fquares of the Szves being in
the fame ratio with the Verfed-fines of the dou-
ble Arches, we may inftead of szyallume for
our Multiplicator est or the Verfed-fine of
3K
4
the
_ ; aren eee de elle
eto. lL pap ictal bw le aca i i al i atl
Mifcellanea Curiofa. 31
the difference of the Latitudes,divided by thrice
the Verfed-fine of the fum of the Co-latitudes,
&c. which is the utmoft Compendium | can think
of fer this purpofe, and the fame /eries will
become, ms .
ee et LP pe ve Ue oe
i as BR SV AT Fh ipe
Hereby we are enabled to eftimate the de-
fault of the method of making the Meridian
line, by the continued addition of the Secants
of equidifferent Arches, which as the difference
of thofe Arehes are finaller, does ftill nearer |
and nearer approach the Truth. Ifweaffume,
as Mr. Wright did, the Arch of one minute to
be Unity, and one minute to be the common
difference of a rank of Arches: It will be in all
cafes, as the Arch of one Minute, to its Chord
:: So the Secant of the middle Latitude, to
the firft ftep of our feries. This by reafon of
the near equality between 2 and 25, which are
to one another in the ratio of Unity to 1—0,
- ©000000035§2566457713, &c. will not differ
_ from the Secant fbut in the ninth Figure; be-
ing lefs than it in that proportion. The next
oe 3 3 és
ftep being st 2s will be equal to the Cube |
of the Secant of the middle Latitude multipli_
ie eee
Ae 555°
ed into ar 0,000000007051 32908715 ;
which therefore unlefs the Secant excéed ten
_ times Radius, can never amount to 1 in the fifth
place. Thefe two fteps fufhce to make the
Meridian Line, or Logarithm Tangent to far
more’places than any Tables of Natural Se-
; cafes,
EM OT er ten
22. M ife ni C fa
cants yet extant, are computed to; butifthe —
third ftep be required, it willbe found tobe
DAC
4-5 into eae 0000000000000000894598 5
By all which it appears, that Mr. Wricht’s
Table does no where exceed the true Meridian
Parts by fully half a Minute: which finall dif-
ference arifes by ES having added continu ually
: the Secants of 1’, 2’, 3’, &c. inftead of 037,12,
2/55 32, &c, But asit is, it is abundantly fuf-—
ficient for Nautical Vfes. hatin Sir Jonas Afoor’s
New Syftem of the Afathematicks, is much near-
-erthe Truth, but the difference from Wright
is fcarce fenfible till you exceed thofe Latitudes
where Navigation ceafes to be practicable, the
one exceeding the Truth by about half a Mi-
nute, the other being a very fmall matter de-
ficient therefrom.
For an Example eafie to be imitated by who-
fo pleafes, I have added the true Meridional
Parts to the firft-and laft Minutes of the Qua-
drant; not fo much that there is any occafion
for fach occurrancy, as to fhew that | have ob-
tained, and laid down herein, the-full Doctrine
of thefe Spiral Rhumbs, which are of fo great
concern inthe Art of Navigation. - :
‘7bne firft Minute is, I 60000001 41036$862178
The Second, §'+.. ©2,00000005541063806707
The Lait or 89° 59 is 30374963431 1414228643
a not 32348, 5279 as Mr. Wright has it, by
adding the Secants of every whole Minute: Nor.
30249,8 as. Mr. Oughtred’s-Rule makes it, by
adding the Secants: of. every other half Minute.
-Not 30364,3 as Sir. Fonas Algor’ had concluded |
. it
— Mifcelanea Cutiofa. 22
it by I know not what Method, tho’ in the reft
of his Table he follows Oughtred.
_ And this may faffice to fhew how to derive
the true Meridian Line from the Sines, Tan-
gents, or Secants fuppofed ready made ; but
we are not deftitute of a Method for deducin
‘the fame independently, from the Arch it felt.
If the Latitude from the Equator be eftimated
by the length of its Arch 4, Radius being U-
nity, and the Arch put for an Jnteger be 4, as
before; the Meridional parts anfwering to that
Latitude, will be .
—intodl3 4 4-34! jase
e4 F\. +e.
7 Aor 5045 4 k 2yie
fA” or aii i Wn.
P jal
which converges'much: fwifter than any of the
former Series, and befides has the advantage of
A encreafing in Arithmetical progreflion,which
would be of great eafe, if any fhould undertake
de novo to make the Logarithm Tangents, or the
Meridian Line to many more places than now
we have them. The Logarithm Tangent to |
the Arch of 45 - 3 A being no other than the
aforefaid Series 4 +4 4? |, A’, &c. in
“Napeir’s form, or the fame multiplied into
0,43429, &c. for Brigg’s. |
_ But becaufe all thefe Serzes toward the latter
€nd of the Quadrant do converge exceeding
flowly, fo as to render this Method almoft ufe-
Tefs, or at leaft very tedious: It will be con-
venient to apply fome other Arts, by afluming
the Secants of fome intermediate Latitudes ;
“and you may for s or the Sine of #2 the Arch of
shalfthe difference of Latitudes, fubftitute -—
ge Tate OM ayes 2 rater &c. accord-
iy ing
pea tat a fe hash Se ab i
ea Chea i ees a ol,
ee
34 Mifcellanea Curiofa.
ing to Mr. Newton's Rule for giving the Sine
from the Arch: And ifabe no more than a
Degree, a very few fteps will fuffice for all the
_ accuracy that canbedefired.
_ And ife be commenfurable to a, that is, if
it be a certain number of thofe Arches with
which you make your é#teger, then will £ be
tate
that number: which if we call 2, tbe parts. of’
the Meridional Line will be found to be, |
(t -|- faa frat es fae, &e.
‘r+ nF 7 rt=
maa fat mm fies, &e
fi. Sr. Gir 6 7.1?
, inte 4 ge 5. 1i3fzas%, Ke.
3 | 12074 J \(960)7* 3}
woe The eg Ate
; 5040 re
L
Inthis,the firft two fteps are generally fufhei-
ent for Nautical ufes,efpecially when neither of
the Latitudes exceed 60 degrees, and the dif-
ference of Latitudes doth not pafs 30 degrees,
But I am fenfible I have already faid too
much for the Learned, tho’ too little for the
‘Learner ; to fuch 1 can recommend no better
-Treatife, than Dr. Wallies precedent Dif
‘courfe, wherein he has with his ufual brevity,
-and that perfpecuity peculiar to himfelf, hand-
led this Subje& from the firft Principles, which
here for the moft part we fuppofe known, _
I need not fhew how, by regreflive work, to
-find the Latitudes from the Meridional Parts.
the Method being fufficiently obvious. 1 shall
-only conclude with the propofal of a aberen
whic
mn
ew
Mifeellanea Curtofa. 25
which remains to make this’Doctrine compleat,
and that is this. - 102 2% f
_ A Ship fails from a given Latitude, and ha-
ving run’a certain number of Leagues, has al-
terd her Longitude by'a givemangle, itis re-
quired to find the Courfe fteared. Thefolu-
tion hereof would be very acceptable, if not
to the Publick, at- leaft to the Author. of this
Tratt, being likely to open fome further Light
into the Myfteries of Geometry. ©
To conclude, I fhall-only addy That Uni- |
ty being Radius, the Co-five of the Arch 4,
according to the fame Rules of Mr. Nemtor,
will be ‘
_
.
to-b A 3 A ngs A bistas A ngs ais3 Ah KO
from which and the former Series exhibiting
the Size by the Arch, by divifion, it is. eafie
to conclude, that the Natural Tanzent of the
Arch 4,is ae
An At Ot Aah 4, &e.
and the Natural Secant to the fame Arch |
ES As At AST A", &e.
and from the Arithmetick of Infinites, the
Number of thefe Secants being the Arch A,
it follows, that the fum Total of all the In-
finite Secants on that Arch, is
| AT; wire vets Ai+yiite Dige XC
Ch > he
a .
= Saat 4 : the
36 Mifcellanea Curtofa.
the which, by what foregoes, is the Logarithm
Tangent of Napeir’s form, for the Arch of 4sgr.
4-1 4, as before. | a
' And Colle@ing the Infinite Sum of all the
Watural Tangents on the faid Arch A, there
williarlid .£91597] 9 y aks : i
pAAt A AME A baits AM its A, Ke.
which will be found to be the Logaritm of the
Secant ofthe fame Arch.
: MifceHanea Curiofa. 27
A moft compendious and facile Me-
- thod for Conftrucing the Loga-
_ rithms exemplified and demonftra-
- ted fromthe Nature of Numbers,
_ without any regard to the Hyper-
bola, with a [peedy Method for
finding the Number from the Lo-
. garithm given. By E. Halley.
“4H & Invention of the Logarithms is juft-
“i ly efteemed one of the moft, Ufeful Dif-
‘coveries in the Art of Numbers, and accord-
ingly has had an Univerfal Reception'‘and Ap-
plaufe; and the great Geometricians of this
Age, have not been wanting to cultivate this
Subje& with all the’ Accuracy and Subtilty a
‘matter of that confequence doth require ; and
‘they have demonftrated feveral very admira-
ble Properties of thefe Artificial Numbers,
‘which have ‘réndred their Conftruétion much
-more facile than by thofe opetofe Methods at
‘firft ufed by thelr truly Noble Inventor, the
Bet any and our worthy Country-man
TRAE Brey p9. 2! © A8% otto oda to radians.) oy
Bat notwithftanding all their Bedeavours, I
“find very fewf thofe: who) make conftant ufe
of Logarithms,’ to: have attained an adequate
Notion ofthem to know how ito make or ex-
amine them 3! 6t-:to- cag thenextent if
3 the
38 Mifcellanea Curifa.
ufe of them: Contenting themfelves with the ©
Tables of them as they find them, without da-
ring to queftion them, or caring to know how —
to réCifie. them, fhould they be found amifs; —
being I fuppofe under the apprehenfion of fome —
creat difficulty therein. +: For.the fake. of fuch
the following Tra is principally intended,but —
not without hopes however to produce fome-
thing that may be acceptableto the moft know-
payee Gel thefe \mattersissreyy wae Ay ae
But firft,it maybe requifite.to premife a de-
finition.of Logarithms, in order to render the
enfuing Difcourfe more clear, the rather be-
caufe the old one Numerorum proportionalium
aqui differentes comites, feems too {canty to de- —
fine them fally. They may more properly be
faid to be Numer: Rationum Exponentes: Where-
in we confider ratio as a Quantitas fui. generis,
beginning ‘from the ‘ratio, of requality, or 1 to
1==0; being, Affirmative when the ratio 1s 1n-
-‘ereafing, as of Unity toa greater Number,
’ “but Negative when -decreafing; and,thefe ra-
tiones we fappofe to be meafured by the Num-
“ber of ratiuncule contained) in each. Now.thefe
-yatiuncula are fo to’be underftood as ina
-continued Scale of Proportionals. infinite -in
‘Number between the two terms of the ratio,
* which infinite Number ofymean Proportionals
‘is to that infinite Number of siete and equal
“yatiuncule betweenvany other two. terms, as
the Logarithm of the one ratio is tothe Loga-
sithm of the other. ‘Thus, if there be fuppo-
* fed’ hetween 1and to an infinite Scale of mean
© Proportionals, whofe Number is, 1©0000, Cc.
~ Gn infinitum 5 between 1 and 2-there -fhall be
30102, -C*c.. of fuch Proportionals,and between
be e tt rast: “f
Mifcelanea Curiofa. 39
1 and 3 there will be 47712 Gc. of them?
which Numbers therefore are the Logarithm’
of the rationes of 1 to 10, 1 to 2, and 1 to 3:
and not fo properly to be called the Logarithms
of 10, 2 and 3. :
_ But if inftead of fuppofing the Logarithms
compofed of a number of equal Ratiuycula,
proportional to each ratio, we fhall 1 take
the ratio of Unity to any number to confift
always of the fame infinite number of Xa-
tiuncula, their magnitude, in this cafe, wil!
be as their number in the former; wherefore
if between Unity and any Number propofed,
there be taken any infinity of mean Proportio-
nals, the infinitely little augment ‘or decre-
ment of the firft of thofe means from Unity,
will be a ratiuucula, that is, the momentum or
Fluxion of the ratio of Unity to the faid Num-
ber: And feeing that in thefe continual Pro-
portionals all theratiuncule are equal,their Sum,
or the whole ratzo will be as the faid momentum
is directly ; that is, the Logarithm of each
yatio will be as the Fluxion thereof. Where-
fore if the Root of any infinite Power be ex-
tracted out of any Number, the differentrola of
the faid Root from Unity, fhall beas the Loga-
rithm of that Number. So that. Logarithms
thus produced may be of as many forms as you
pleafe to affume infinite Izdices of the Power
whofe Root you feek: as if the Index be fuppo-
‘fed 100000@c. infinitely, the Roots fhall be the
‘Logarithms invented by the Lord Napeir 5, but
af the faid Index were 2302585, @c.Mr.Briggs’s
‘Logarithms would immediately be produced.
And if you pleafe to ftop at any number of
Figures, and not to continue them on, it will
Cs es ama Ta See Soe
40 Muifcellanea Curiofa.
fuffice to affume an- Index of a Figure or two
more than your intended Logarithm is to have, —
as Mr. Briggs did, who to have his Logarithms —
true to 14 places, by continual extraGtion of
the Square Root, at laft came to have the Root
of the 140737488355328rh Power; but how
operofe that ExtraGtion was,will be eafily judg-—
ed by whofo fhall undertake to examine his
— Calculus. ie
Now, though the Notion of an Infinite Pow-
er may feem very ftrange, and to thofe that
know the difficulty of the Extraction of the
Roots of High Powers, perhaps impracticable ;
yet by the help of that admirable Invention of
Mr. Newton, whereby he determines the Uncie
or Numbers prefix’d to the Members compo-
fing Powers ‘(on which chiefly depends the Do-
étrine of Series) the Infinity of the Index con-
tributes to. render the Expreffion much more
eafie: For if the Infinite Power to be refol-
ved be put (after Mr. Newron’s Method )
I * TEMS aa f pe at
Pipa Pepg lm 8 Tam inftead of Ind Fr
in |) amebomm , | 1—6m-\1 rmm—6m3 ©
2mm qq Tse ee 240k. g
&c. (which isthe Root when is finité)becomes
gh le ee
oh ee Ooo sp Tamme Be maaan: Sith
mm being imfimte infinite, and confequently
whatever is divided thereby vanifhing. Hence
it follows that ae multiplied. into 9-2 994-3
gqq—. 4+ 49° &c. is the augment of the firft
of our mean Proportionals between Unity and
1+}-q, and is therefore the Logarithm of the
ratio of 1 to 1-+q; and whereas the Infinite bie
Yr
— Mifcellanea Curiofa. 41
dex m may be taken at pleafure, the feveral
Scales of Logarithms to fuch Jzdices will be as
—or reciprocally as the Indices. And if the
idee be: taken 10000, &c. asin the cafe of
Napeir’s Logarithms, they will be dimply ie.
9903 999--; F151 29° &e
Again, if the Logarithm of a Secteatiie raq-
tio be “pagel the infinite Root of Iq or
3 i"
pe =7% &c. whence the. decrement of the
fir ft oe infinite Number of Propor ian
will be = into ght agjigitigt+igttig®
which therefore will be.as the shing ’
the ratio of Unity tor—g. But if m be put
"19000, &c: then the faid Logarithm will be
qiaqtig igi ts 9° Taq’, &c.
_. _ Hence the ferns of any ratio, heing 4 and é b,
“gq becomes ~ - or the difference divided by
the. leffer term, when ’ tis an increafing _ ratio;
or, h @ when ’tis decreafi ing, or as b to 4.
“Whence the Logarithm. of the fame ratio may
_be doubly expreit, for putting ~ for the diffe-
jade of the ter ms 4 and. b, it will be either.
eas peat
x2 x3 xt xs 36
aa, OA Ly eae ae CXC.
240 3 43 ree abs as
But
rai into = —
A? Mifcellanea Curiofa.
But if the ratio of ato b be fuppofed divi-
ded into two parts, viz. into the ratio of 2 to_
the Arithmetical Mean between the terms,
and the ratio of the faid Arithmetical Mean
to the other term 6, then will the Sum of the
Logarithms of thofetwo ratioves be the Loga-
rithm of the ratio of a to. 6; and fubftituting
+zinftead of }4‘+3 6 the faid Arithmetical
Mean, the Logarithms of thofe ratzones will be
by the foregoing Rule, |
© ews OG we CS Fe x5
= —s
ea) eee etree
® in oe ene eee —-——-— ‘Sr and me
wm errrarr area lrcar sr: re |
SES ge) gee Sigg a) aeB i Deg ST nat gee
YQ ee | &
ip tk Sh re avila 6.2% Cae
the Sum!._ 2x 2x3 2x5 Ox? As
. bethe Logarithm of the ratio of 4 to b, whofe
‘difference is x and Sumz. And this ‘Series
converges twice as fwift as the former, and
therefore 1s more proper for the Practice of
making Logarithms: Which it performs with
that expedition, that where» the difference
is but the hundredth part of the Sum, the
firft ftep == fuffices to feven places of the
Logarithm, and the fecond ftep to twelve:
But if Briggs’s firft Twenty Chiliads of Loga-
‘rithms be fuppofed made, as he has very care-
fully computed them to fourteen places, the
firft ftep alone, is capable to give the Loga-
rithm of any intermediate Number true; to. all
the places of thofe Tables. :
After the fame manner may the difference
of the-faid two Eogarithms be very fitly ap-
: | bapa
,
Maifeellanea Curtofa. . An
plied to.find the. Logarithms of Prime Num-
bers, having the Logarithms of the two next
Numbers above and below them: For the dif-
ference of the ratio of ato + z and of yz tob
is, the ratio of 2.6 to} xz, and the half of that
ratio, is that of Vabto ! z, or of the Geo-
metrical Mean to the Arithmetical. And
confequently the Logarithm thereof will be the
half difference of the Logarithms of thofe-rati-
ONES, Vie!» eid
Ge CONS ae ‘
x4 pia i, by
— into — + — te AE 2
2h 4rt } 62° 8z8
Which is a Theorem of good difpatch to find
_ the-Logarithm of 3 z. But the fame is yet much
more advantageoufly performed by a Rule de-
rived from the foregoing, and. beyond which, |
in-my-Opinion, nothing better.can be hoped.
For the ratio of abtoz zzor', aattab+%
bb, has the difference of its terms % 2a—? ab-}-
4, bb, or the Square of 3 4—5 b—} x x, which
in the prefent cafe of finding the Logarithms
-of Prime Numbers;.is always Unity, and cal-
ding tbe Sum of the terms 4 zz -|-4 b==yy, the
-Logarithm.of the ratio of / 2b to z4--3b or}
_~ will be found
Mt eg ty iB i I :
eee Cale oie rest era &e. >
“which ‘converges very much fafter than anly
\ Theorem hitherto publithed for this purpofe-
“Here note. ~ is all along applied to adapt thefe
) aid |
eye eT
re et ORIN. LOO tod 9: ' it Rules
44. Mifcellanea Curiofa,
Rules to all forts. of Logarithms. If m be
rocco &c. it may be neglected, and you will
have JVapeir’s Logarithms, as was hinted be- _
fore; but ifyou defire Briggs’s Logarithms,
which are now . general]
divide your Series by |.
Ret “TRO
rae at how 3
y received, you muft -
2,302585092994045684017991 454684364207
_ 60110148862377297603 3328 Dae
or multiply it by the reciprocal thereof, #7z.
©543429448190324518276511280189166050822
94397003 803666 566114454-- 6)
But to fave fo operofe a Multiplication
(which is more than all the reft of the Work)
‘itis expedient to divide this Multiplicator by
the Powers of z or y continually, according to
the Direction of the Theorem, efpecially where
x 1s {mall and Integer, ‘referving the proper
Quotes to be added together, when you have
produced your Logarithm to-as many Figures
as you defire: Of which Method fF will give
a Specimen. fit 20 tes Lot gaey aay i
If the Curiofity of any Gentleman that has
leiftre, would prompt him to undertake todo
the Logarithms of all Prime Numbers under
¥00000, to 25 or 30 Figures, I dare affure him,
that the facility of this Method willinvite him
thereto 5 nor can any. thing more eafie be de-
fired. And to encourage him, £ here give the
_ Logarithms of the firft Prime Numbers under
36 to 60 places, computed:by-the accurate Pen
of Mr. Abraham Sharp, (from- whofe Induftry
and Capacity the World may .in time expect
great Performances) as they were ‘communi-
“cated to me by our common Friend Mr. Euclid
Speidall. !
Numb.
— Mifeellanea Curiofa. 4s
Numb. Logar ithm.
2 0,30102999566398119521 373889472449
3026768189881 462108541310427
B Os47'7121254.719662.43729502790325511
§ 3092001 283641 90695864329866
J 0584509804001 425683071 221625859263
«61 93.4835 7239632396 5406 503835
TL 150413926851 §822504075019997124302
} 42.4.1'706 70219046645 3094596539.
13 1,11 3943352306837769206 54189502624
* "6 254561189005053673288598083 |
17 1 52304.48921 3782730285 4016989432833
aif "1030007567378425046397380368
AQ 1,27875360095282896153633347575692
9317951 129337394497598900819
‘The next Prime Rivgihee: IS '23; which. I will
cake for an Example of the foregoing Dottrin,
and. by the firft Rules, the Logarithm of the
ratio of 22 to 23, will be pane to be either
iG 2 SS espagnol bee teor
aes
I I
os Be i I 119364 32181715
Bec.
“As likewife that of the ratio of 23 to 24 by
a like Procefs.
{ T ! ee + Bec
E a
ait aan! area t ssi
&e. - And
46 Mifcellanea Curiofa.
And this is the Refult of the Dodrine of |
Mercator, as improved by the Learned Dr.
cee But by the fecond Theorem, wz. —
ee sage &c. The fame Logarithms
327
a obtained gy ‘eye fteps. To wit, ©
Zag SEO Ca
45 ' 273375 | 922640625 ' 2615686171875
&c. and : | BoA ars
2. a5
ee a Te DT) TLL eee bee ee ness &C.
47 31 1469 ° 1146725035." 3546361343241
which was invented and demonftrated in the
Hyperbolick Spaces Analogous to the Loga- —
rithms, by the Excellent Mr. fames Gregory,
in his Exercitationes Geometrice, and fince fur-
ther profecuted by the aforefaid Mr. Speidall,
in a late Treatife, in Englifh, by him publifhed
on this Subjeé&. But the Demonftration as
I conceive, was never till now perfe&ed with-
out the confideration of the Hyperbola, which
_ ina matter purely Arithmetical as this is, can-
not be fo properly applied. . But what follows ~
Ithink 1 may more juftly claim as my Own, viz.
‘That the Logarithm of the rae/o of the Geo-
metrical Mean to the Arithmetical between
22 and.24, or of / 528 to 23 will be foand t to
be either.
ee | PB ll a oe arava ft jie hos ee
1058 ma ibeea + 395513 Sake 516487882248
&e. or
10§7 | 3542796379 1 $9676 58485285
- All
— Maifcellanea Curiofa. 47
All theife Series being to be multiplied into
044342944819 &c. if you defign to make the
- Logarithm of Briges. But with great Advan-
tage in refpect of the Work; the faid 434294
4819, &c. is divided by 1057 and: the Quo-
tient thereof again divided by three times the
Square of 1057, and that Quotient again by 4
of that Square, and that Quotient by ? there-
‘of, and fo forth, till you have as many Figures
‘of your Logarithm as you defire. As for Ex-
ample; the Logarithm of the Geometrical
Mean, between 22 and 24, is found by the Lo-
garithms of 2, 3 and 11 to be | |
| 10§7)43429 &c.
3 in 1117249)41087 &c.
4 in '1117249)12258 &c.
Zin 1117249)65§832 &c.
3 in 1117249)42088 &c.
1.361 3169612669061 294500917 2669805
( 41087462810146814347315886368
Sede 12258521 544181829460074
i heat 6583 235184376175
iy ee nk"
(
4208829765
Eig
Summa.
| 1.361 7278360175928788677771 1225117
“Which is the Logarithm of 23 to thirty two
places, and obtained by five Divifions with ve-
ry finall Drvifors, all which is much lefs Work
than fimply multiplying the Series into the faid
' -Multiplicator 43429, &c. i |
' Before I pafS on to the converfe of this Pro-
“blem, or to fhew how to find the Number ap-
pertaining
43 Mifcellanea Curwofa.
pertaining to a Logarithm afligned, it will be
requifite to advertife the Reader, that there
is a fmall miftake in the aforefaid Mr. ames
Gregory's Vera Quadratura Circuli,& Hyperbole,
publifhed at Padua Anno 1667. wherein he ap-
plies his ‘Quadrature of the Hyperbola to the
making the Logarithms; In pag. 48. he gives
the Computation of the.Lord Napew’s Loga-
rithm of 10, to five and twenty places, and ©
finds it. 230258509299404.5624017870 inftead
Of 2302585092994045684017991,. erring inthe
eighteenth Figure, as I was. aflured upon my
own Examination of the Number L. here. give
you, and by comparifon thereof with the fame
wrought by another hand, agreeing therewith
to 57 of the 6o-places. Being defirous to be
fitisfied how. this difference arofe, I took the
- no fimall trouble of Examining Mr. Gregory’s
Work, and at length found, that in the infcri-
bed: Polygon of 512. Sides, 5 in the eighteenth
“Figure, was.a o inftead of 9, which being re-
€tified, and the fubfequent _Work corrected
therefrom, the refult did agree to a Unite
-with our Number. - And this I propofe not to
‘Cavil at an eafie miftake in managing of fo va
“Numbers, efpecially by a Hand that has fo
well deferved of the Mathematical Sciences,
~“but.to-fhew'the exad& comcidence of two fo
very differing Methods to make Logarithms,
yaar ‘Might otherwife have’ been. queftion-
~:°Ftom the Logarithm given to find. what
ratio it expreffes, is a Broblem..that has not
been fo much confidered.as the former, but
- whichis. folved:with-the like eafe, and demon-
a Mifcellanea.Curiofa. Ag
ftrated by a like Procefs, from the fame gene-
neral Theorem of Mr. Zvewton: For as the Lo- |.
garithm of theratio of 1to1--7 was proved to be
zs : )
19
be Ae Y fo the Logarithm, which we
will from henceforth call L, being given, 1--L,
will be equal to itg\™ in the one cafe; and
1-—L will be equal to al inthe other: Con-
(my, and that of the ratio of 1 to 1—7 to
24
fequently 75-1)” will be equal to 1-19, and
ae to 1—9; that is, according to Mr.
“Newton’s faid Rule,. 1-|-mL-4~3 7m’ L?-|-4m3 L?--
tims -\-4.m°L? &c: will be==1-4-9, and 1—
mL 4-3m*L? 2m L34- 4m? LAA, 3, mL? &e.
will be equal to 1-9, m being any infinite In-
‘dex whatfoever, which is a full and general
Propofition from the Logarithm given to find
the Number, be the Species of Logarithm what
it will. But if MNapeir’s Logarithm be given,
‘the Multiplication by m is faved (which Mul-
fiplication is indeed no other than the redu-
Pe ocre other Species to his) and the Series will
be more fimple, vz. 1-|-L-}$LL43 L4--5L*4-
Mey &cl of rol Lb shag $b sh
4 €. This Series, efpecially in great Numbers
converges fo flowly, that it were to be withed -
it could be contracted.
3
a
mr
ms} OY 7 :
50 Mifcellanea Curiofa.
‘If one term of the ratio, whereof L is the
Logarithm, be given, the other term will be.
eafily had by the fame Rule: For if L were .
_ greater than the given L, and the difference ~
=
Naperr’s Logarithm of the ratio of athe lefler
to b the greater term, 0 would be the Produ& -
of a into 1-pL-+$LL4"LLL &e. =4paL4
aLL\eaL? &c. But if” were given, 2 would
“be=?—bL-;-46LL—16L? &c. Whence, by the
help of the Chiliads, the Number appertaining
to any Logarithm will be exactly had to the ©
utmoft extent of the Tables. If you feek the
neareft next Logarithm,whether greater or lef- |
fer, and call its Number a if lefler, or 0 if
thereof from the faid neareft Logarithm you
call /; it will follow, that the Number an-
fwering to the Logarithm L will be either a ©
into. 1-}-/- 51 gl El’ &e. or elfe b
into 1—/-{|-4ll—s1L-b + ,75/* &c. wherein as
his lefs, the Serzes will converge the fwifter. |
And if the firft 20000 Logarithms be given to ©
fourteen Places, there is rarely occafion for
the three firft fteps of this Serzes to find the
Number to as many places. But for Ulacq’s.
great Canon of 100000 Logarithms, which is:
made but to ten places, there is fcarce ever.
need for morethan the firft ftep 4-| a/ or a--
malin one cafe, or'elfe b—b1 or b—m b/ in
the other, to have the Number true to as ma-_
ny Figures as thofe Logarithms confift of.
If future Induftry fhall ever produce Loga-
rithmick Tables to. many more places than.
now we have them; the aforefaid Theorems.
will be of more ufe to reduce the correfpondent
Natural Numbers to all the places thereof.
In-order to make the frit Chiliad, ferve all
‘Utes,
— Maifcellanea Curiofa. 51.
Jfes, I was defirous to contra this Series,
wherein all the powers of J are prefent, into
fone, wherein each alternate Power might be
wanting ; but found it neither fo fimple or
uniform as the other. Yet the firft ftep there-
f is, I conceive, moft commodious for Prac-
tice, and withal exa@ enough for. Numbers
not exceeding fourteen places, fuch as are
Mir. Brigs’s large Table of Logarithms; and
herefore I recommend it to common Ufe.
b> It is thus 2 a-- Go! or b— sel
i} | oe eer:
vill be the Number anfwering to the Loga-
ithm given, differing from the Truth by but
me half of the third ftep of the former Series.
ut that which renders it yet more eligible,
that with equal facility, it ferves for
wg’s Or any other fort of Logarithm, with
7 ‘a uae oe ee T ° -
he Only variation of writing — inftead of 1,
: at is, a-|-
a : b! 1 i.
— and b— oe gets
aa ae ——— and
2
i. |
? which are eafily refolv’d into A-
As 43429 &¢e.—2/ to 43420+-5/ : -1 to the
So is 2 { Num-
43429 &e.|-3! to 43429— $12: ae
E 2 If
CE ar i Lys
‘ vale ies bs decid BB i Bey ye
52 Mifcellanea Curiofa.
If more fteps of this Series be defired, it
Ls Me ‘ al 4.al>
will be found as follows, a ieee a
1 als ; ‘ , id | ;
Saey &c. as may eafily be demonftrated by
working out the Divifions in each ftep,
and collecting the Quotes, whofe Sum will be
found to agreee with our former Series.
Thus I hope, I have cleared up the Doc-
trine of Logarithms, and fhewn their Con-
ftruction and Ufe independant from the Ay-
perbola, whofe Affections have hitherto been
made ufe of for this purpofe, though this be a
matter purely Arithmetical, nor properly
demonftrable from the Principles of Geome-
try. Nor have I been obliged to have re-
courfe to the Method of Indivifibles, or the
Arithmetick of Infinites, the whole being no
other than an eafie Corollary to Mr. Newton’s
General Theorem for forming Roots and
Powers. : | rs
| Mifcelanea Curtofa. 53
A SOLUTION,
Given ty Mr. ‘fohn Collins, of a Choro-
_ graphical Problem, Propofed by
Richard yoraicy, Eiq;
PROBLEM.
TheDiftances ofthree Objectsin the ©
fame Plain being given, as 4,B,
C; The Angles made at a fourth
e Place in the fame Plain as at
8, are obferved : The Diftances
from the Place of Obfervation
to the refpective eae are re-
quired.
The Problem hath fix Cafes.
Caf 1. ¥ F the Station be taken without the
Triangle made by the Objects but
inone of the fides thereof produced, as at Sin
the orh Figure; find the Angle ACB; then
in the Triangle "ACS all the Angles and the
fide AC are known, whence either or both the
Diftances SA or SC ie be found.
ae Bs Gea
5 4. Maifcellanea Corfe | |
Cafe 2. If the Station be in one of the Sides
of the Tt iangle, as in the roth Figure at Sy. }
then having the three fides AC, CB, BA given,
find the Angle C_4 B; then again in the Tri-”
angle SA B, all the Angles, and the fide AB,
are known; ence may be found either 4S,
or SB, Geometrically, if you make the “Angie.
CAD equal to the obferved Angle c S B, and
draw BS parallel to DA, you determine the
Point of Station S.
Cafe 3.1f the three Obj edtslie in a right Ling!
as ACB (fuppofe it fone and that a Circle
pafleth through the Station S, and thetwo ex-
terior Objects A B, then is the Angle ABD
equal to the obferved Augle A sc. (by 21 of
_ the 3d Book of Euclid) as infifting on the fame
Arch AD: And the Angle BAD in like
manner equal to the obferved Angle c SB:
By this means, the point D is determined.
Join DC, and produce the fame, thn a Circle
pafling through Points A B D, interfedts Dc,
: produced at S, the place of Station.
Calculation.
In the Triangle 4 B D, all the ovine: and
the fide 4 Bare known, whence may be i
the fide 4 D.
Then in the Triangle C_A D the two fides
c A and AD are known and their contained
Atigle c AD is known ; whence may be found
the Angles cD A and ACD, the comple-
_ ment whereof to a Semicircle is the Angle $
cA: in which Triangle the Angles are now
all known and the fide 4C: whence may be
found either of the Diftances, SC or Sale
| Cafe +
WF ee ees Sy
mah x van
: Mifcellanea Curiofa. | 55
angle, made by the Objects, the fum of the
Angles obferved is lefs than four right Angles.
The Conftruction is the fame as in the laft
Cafe, and the Calculation likewife ; faving
that you muft make one Operation more, ha-
ving the three Sides, 4C, C B, BA, thereby
find the Angle cA B, which add to the Angle
E AD, then you have the two fides, viz. A
C, being one of the Diftances,and 4 D, (found
as in the former Cafe) with their contained
Angle C_AD, given to find the Angles CDA,
and 4 C D, the Complement whereof to 4a Se-
micircle, is the Angle SC 4: Now inthe Tri-
angle Sc.4, the Angle at c being found, and
at S obferved, and given by Suppofition, the
other at 4 is likewife known, as being the
complementof the two former to a Semicircle,
and the fide 4 C given ; hence the Diftancec S
or 4S may be found.
- Cafe 5. If the place of Station be at. fome
Point within the plain of the Triangle, made
by the three Objects, the Conftruction and
Calculation is the fame as in the laft, faving
only that inftead of the obferved Angle 4 Sc,
the Angle 4 B Dis equal to the Complement
thereof to a Semicircle, to wit, it is equal to
the Angle 4 SD; both of them infifting.on the
fame Arch 4D: And in like manner the An-
sie B 4D is equal tothe Angle DS B, which
is the Complement of the obferved C S B; and
in this Cafe, the. fum of the three Angles ob-
ferved,: 1s equal to four right Angles.
B 3 In
Cafe 4. If the Station be without the Tri-
‘
Biss aie tS gat
56 Mi fc dee Curio ip a.
In thefe three latter Cafes no ufe is oN of |
the Angle obferved.between the two Objects, ©
as 4 and B, that are made the Bafe-line of the
Conftr uétion : ; yet the fame is of ready ufe for /
finding the third Diftance, or laft fide fought,
as in the fourth Scheme, in ‘the Triangle S 4 2B,
theré is given the Diftance 4 B, its oppofite-
Angte equal to the fum of the two obferved
‘Angles, and the An@le S 4B attained, as in
the fourth Cafe: Hence the third Side or laf
Diftance SB may be found.
And here it may be noted, that the three
Angles C 4S, 4S 8B, SBC, are together equal
to the Angle ACB, "for the two Angles c SB,
and Cc BS, are equal to ECB, as being the
Complement of SC B to two right Angles; and
the like in the Triangle on the other em
EPS OPO ET
Cale G2 IE the diget Objeas be' 4, B,C, anid
the Station at S, as before, it may happen, ac=
‘cording to the former Conftructions, that the
Points C and D may fall clofe together, and
fo a right Line joining them may be produced
with uncertainty : ; in fach cafe the Circle may
be conceived to pafs through the place of Sta-
tion at S, and any two of the Objects (as im
the fixth’ Scheme) through Band C; wherein
making the Angle DBC équal to the obferved
Angle A Sic} and Bt D equal to the Comple-
tment to 1 80 degrees of both the obferved An-
gles in DSB thereby the Point D is determi-
ned, through which, and the points cB, the
Circle isto be defcribed , and joyning D 4,
(produced, when 5 requireth) where it in-
BP: asic:
a
i
1
%
ce em
SR a ea a j
K Plater, pape 87
Cine:
i Mifeellanea Curiofa. Hee 37
Be eitetts, the Circle, as at S, is the place of Sta-
coe fought.
ig This Problem may be of oond Ufe for the
_ due Situation of Sands or Rocks, that are
within fight of three Places upon Land, whofe
diftances are well known; or for chorographi-
cal Ules, &c. Efpecially now there isa Me-
- thod of obfervingAngles nicely accurate by aid
of the Telefcope ; and was therefore thought fit
to be now Publifh’d though it be a competent
time fince it was delivered in Writing.
The
‘ad ha "s TE ETE RT a ae ee
¥ : s + = ae ay ee
j A i
a Maifcelanea Curiofa. ae oe
| The Solutions of three Chor oer aphic
- Problems, by a Member of the
Philofophical Society of Ox-
ford. ce
HE three following Problems may
occur at Sea, in findimg the diltance
and pofition of Rocks, Sands, &c. from the Sea
Shoar; or in the Surveying of the Sea Coaft ;
When only two Objects, whofe diftance from
each other is known, can be feen at one Sta-
tion ; but efpecially they may be ufeful to one
that would make a JZap of a Country bya
Series of Triangles derived from one or more
~meafured Bafes ; which is the moft.exact way
of finding the bearing and diftance of Places —
from each other, and thence their true Lon-
gitude and Latitude; and may confequently
occur to one that would in that manner mea-
fure a Degree on the Earth. vs
The firft Problem (Fig. 3 and 4.)
There are two Objects, B and C, whofe di-
ftance BC is known; and there are two fta-
tions at .4 and E, where the Objetts Bc
being vifible, and the Stations one from
another, the Angles BaC, B4AE, AES,
A Ec, are known by Obfervation, (which
may be made with an ordinary Serveying Se-
micircle, or Croftaf, or if the Objects be
beyond
Mifcellanea Curiofa. 9.
beyond the view of the naked Eye, with a
- Telefeopick, Quadrant) to find the diftances or
lines AB, AC, AE, EC. |
ee Conftrution.
In each of the Triangles BAE, CAE,
two Angles at -4, E, being known, the third
is alfo known: then take any line #¢ at plea-
fure, on which conftitute the Triangles Bas,
a¢y refpectively equiangular to the Triangles
BAE, AEC; join8y. Then upon BC con-
ftitute the Triangles BC A, BCE, equiangu-
lar to the correfpondent triangles By, By,
_ join AE, and the thing is manifeftly doge. ©
Tie Calculation.
Affuming #¢ of any number of parts, in
triangles « B¢, ey, the angles being given,
the fides «8, “¥,¢8,2y may be found by
Trignometry: Then in the Triangle pay,
having the angle 824, and the legs af, 7,
“wemay find @y. Then By. BC:: 2a, BA:
Pees ya Cas yt Cr.
The fecond Problem (Fig. § and 6.)
_ Three Objedts B, C, D, are given, or (which
is the fame) the fides, and confequently an-
sles of the triangle BCD are given; alfo
there are two points or ftations A, E, fuch,
that at 4 may be feen the three points BCE,
but not D; and at the ftation E may be
feen .4,C,D, but not B, that is the angles
me PAC, BAE, AEC, AED, (and confe-
quently E AC, AEC, are known by obferva-
Ween; co find the lines 48, 0, AE EC,
L ED. Con-
60 = Mifcellanea Curiofa.
— Conftruttion. gic
Take any line ¢ at pleafure, and at its ex-
tremities make the angles «ey, #8, aeyz,
aed equal to the correfpondent obferved an-
Bless EAC, EAB, AEC, AED. Produce & 2,
#<, till they meet in?, joing»; then upon
CB deferibe (according to 33. 3. Eucl.) a feg-
ment of a circle that may contain an angle=
¥¢B6; and upon CD defcribea feoment of a
Circle capable of an angle=7 9S; fuppofe F
the common fection of thefe two circles ; join
FB, FC, FD; then from the point C, draw
forth the lines C_4, CE, fo that the angle FCA
may be==? 7 2, and FCE==97¢; fo A, E, the
common Sections of CA, CE; with FB, FD,
will be the points required, from whence the
reft is eafily deduced.
The Calculation.
Affuming «¢ of any number, in the trian-
gles yt, #%¢, all the angles being given, ~ |
with the fide #¢ aflum’d, the fides « >, ¢ ¥, ?,
29, will be known; then in the triangle 7 29,
the angle 7 #?, with the legse7, +9, being
known, the angles «97, 279, with the fide
®y will be known: then as for the reft of the
work in the other figure, the triangle BC D
having all its fides and angles.known, and the ~
angles BFC, BF D, being equal to the found
Bey, 894; how to find FB, FC, FD by Cal-
Culation (and alfo Protrattion) is fhewn by Mr.
Collizs in the precedent Difcourfe, as to all
its cafes, which may therefore fuperfede my
fhewing any other way. ae ne
But
Mifcellanea Curiofa: 64
But here it muft be noted, that if the fim
of che obferved angles, B AE, AED, is i180
degrees : then 42 and E D cannot meet, be-
caufe they are parallel, and confequently the
given Solution cannot, take place ; for which
reafon I here fubjoin another.
Another Solution me |
Upon Bc (Fig. 7.) defcribe afegment BAC
of a circle, fo that the angle of the fegment
may be equal to the obferved 4827, (which
as above quoted is fhewn 33. 3. Euclid.) and
upon CD defcribe a fegment C E D of a circle
capable of anangle equal to the obferved CED;
from c draw the diameters of thefe circles cG
cH; then upon cG defcribe a fegment of a
circle G FC, capable of an angle equal to the
obferved L AEC; likewife upon c Hdefcribe
a circle’s fegment c FH, capable of anangle
% equal to the obferved c 4E: fuppofe F the
common Seétion of the two laft circles HF c,
GFC, join FH, cutting the circle HEC in E,
join alfo FG, cutting the circleG AC ina: }
fay that 4, Z, are the points required.
Demonftration.
“For the LB 4C is=gay by conftruction of
the fegment, alfo the angles cE H, C 4G,
are right, becaufe each exifts in a femicircle:
- therefore a circle being defcribed upon C F as
a.diameter, will pafs through E, 4; There-
fore the angle C4 E==-L cFE=C pi=(by
conftruction) to the obferved angley¢*. In
like manner the LC E A==c F 4c FG= ob-
ferv’d qarte y 6 te
In the ftations 4, F, fallin a right line with
the point C; the lines G 4, HE being paral-
i Sie | | le},
62 Mifcellanea Curiofa.
lel, cannot meet: but in this cafe the Problem
is indeterminate and capable of infinite Solu-.
tions. For as before upon CG defcribe a Seg-
ment ofa circle capable of the obferved Ly: «,
and upon C H, defcribe a Segment capable of —
the obferved 72¢: then through c, draw a
line any way cutting the circles in 4, E, thefe
points willanfwer the queftion. |
The third Problem.
Four points B, C,D,F, (Fig. 8.) or the 4 fides
of a quadrilateral, with the angles compre-
hended are given; alfo there are two ftati-
ons:4 and E fuch, that at-4, only 8.0 2 are ~
vifible, and at E only ADF, that is, the
angles BAC, BAE, AED, DEF are given:
to find the places of the two poiuts 4£, and
confequently, the lengths ofthe lines 4B, AC,
AE, ED, EF. :
Conftruttion. a)
Upon BC (by 33. 3. Eucl.) defcribe a feg-
ment of a circle, that may contain an angle
equal to the obferved angle BAC, then from
C draw the Chord C 4%, or a line cutting the —
circle in 77; fo that the Angle BCA4 may
be equal to the fupplement of the obferved
angle BAE, 1. €. its refidue to 180 degrees.
In like manner on D F defcribe a fegment of
a circle, capable of an angle equal tothe ob-
ferv’'d DE F,and from D draw the Chord DN,
fo that the angle F D N may be equal to the
fupplement of the obferv’d angle 4 £F, join
MN, cutting the two circles in 4,£: I fay —
/, ®, are the two points requir’d.
Demone
—Mifeelanea Curwfa, 62
7 | Demonftration. aed
Join 48,40, ED, £F, thenisthe £ ACAB
LBC (by 21.3. Eucl.)= fupplement of
the obfervd 4 8 4 E by conftrution, therefore
the conftructed 4 8 4 £, is equal to that which ©
was obferved. Alfo the 4 BaCof the feg-
- ment is the’conftruction of the Segment, e-.
' qualtotheobferv’'d £3.4c. In like manner
the conftruéted angles 4EF, and DEF,-are
equal to the correfpondent obferved angles
AEF, DEF, therefore 4£ are the points re-
guir’d. Pith
‘ The Calculation: |
_- In the Triangle BC AZ, the L Bc 24 (=fup-
_ plement of B4£) and 4B AIC (= BAC) are
_ given, with the fide Bc ; thence A¢C may be
found; in like manner DNinthe ADNF
may be found. But the 4 AfcD)=3 cD---
- pC M)is known,with its legs 74.c, c D> there-
fore its Bafe A7D, and LA7DC, may be
known. Thereforethe LDN (=cDF.-
CD M--F DN) is known, with its legs Mp,
DWN; thence MN with the angles DMN,
DNM, willbe known Then the4cMA(=
(LDMC --D MN) is known, with the MAC
==MAB+B4AC): and MC before found ; there-
fore MA and 4c will be known. In like
manner in the triangle E DN, theangles £, Nn,
- with the fide D N-being known, the fides EN,
ED, will] be known; therefore 4 E (= M N---
_ M4A--EN) is known. Alfo in the triangle 4
ec, the 4 A with its fides BC, C 4, being
- known, the fide 4 2B, will be known, with the -
£8c 4310 inthe triangle £ FD, the 4 £ with
- the fides, ED, DF being known, EF will be
found, with the LEDF. Laftly, in the Fas
gle
OE a aT eRe 4
64 Maifcellanea Curiofa:
gle Ac D, the L AC D (SBC D--BC 4) with
its legs 4C, c D being known, the fide 4 D,
will be known; and in like manner EC in the
triangle EDC. ae
Note, that in this Problem, as alfo in the
firft and fecond, ifthe two ftations fallin a
right line with either of the given Objeds:
the locus of 4, or E, being a circle, the par-
ticular point of 4, or E, cannot be determined ©
from the things given. ae
Asto the other cafes of this third Problem,
wherein 4 and £, may fhift places, 7.e. only
_DFE, may be vifible at 4, and only 4, B,c,
at E; or wherein 8, .D,E, may be vifible at
A, and only c, F, 4, at E; or wherein 4 may
be of one fide of the quadrilateral, and E on
the other; or one of the ftations within the
quadrilateral, and the other without it: I
fhall for brevity fake omit the Figures, and
diverfity of the Sines ~- and-- in the caleula-
tion, and prefume that the Surveyor will ea- -
fily direé&t himfelf in thofe cafes, by what has
been faid. ly th
The folution of this third Problem is gene-
ral, and ferves alfo for both the precedent.
For fuppofe ¢ D, the fame point in the laft f-
gure, and it gives the folution of the fecond —
Problem: but if BC be fuppos’d the fame
points with D, F, by proceeding as in the laft,
you may directly folve the firft Problem.
a 7 ae Tea
+ Fe eae eng
Mifeelanea Curtofa. 65
Aa Arithmetical Par ‘ad ox, concern-.
tg the Chances of Lotteries ; by
aie Honouradle Francis Roberts,
ere Fellow of the R. 5.
S fome Truths (like the Axioms of Geo-
metry and Afetaphyficks) are felf-evident
at the firft View, fo there are others no lefs
certain in their Foundation, that have a very
different Afpect, and without a ftri@ and care-
‘ap Examination, rather feem repugnant.
We may find Inftances of this kind in moft
‘Sciences. t
_ In Geometry, That a Body of an infinite
poet, may yet have but a finite Magni-
tude ¢
In Geography, That if Axtwerp be due Eaft
to London, for that reafon London cannot be
Weft to Amvwerp.
In Affronomy, That at the Bardadoes (and
‘other Places between the Line and Tropick)
‘the Sun, part of the Year, comes twice ina
Morning to fome Points of the Compafs.
Jn Hydroftaricks, That a hollow. Cone (ftand-
‘ing upon its Bafis) being fill’d with Water,
the Water hhall prefs the bottom with three
times the Weight, as if the fame Water was .
frozenito Ice ; and Figures might be contriv’d
to make it aha a hundred times as much.
, SaaS j ’
. nat Bt; The
¥4
See ars ae
| ”
66 Mifcellanea Curtofa.
’ Thefe Speculations, as they are generally —
pleafant, fo they may alfo be of good ufe to
warn us of the Miftakes we are liable to, by
carelefs and fuperficial Reafoning. ns
I fhall add one Inftance in Arithmetick,which
perhaps may feem as great a Paradox as any
of the former. ?
There are two Lotteries, at either of which
a Gamefter paying a Shilling for a Lot or
Throw; The firft Lottery upon a juft Com-
putation of the Odds, has 3 to 1 of the Game-
{ter ; the Second Lottery, but 2 to one ; ne-
verthelefs, the Gamefter has the very fame
difadvantage (and no more) in playing at the
Firft Lottery, asthe Second. 3
It looks very like a Contradiétion, that the:
Difadvantage fhould be no greater in playing
againft 3 to 1, than 2 to1, but it may thus be:
refolv’d. . | i
io)
a € f-) ps S70 . 16 pence
ie 8s 4 a-piece.
a) ° ie ;
wo \- 8 ip. a yas a, ©2 fhilling ogee
In the firft Lottery the Gamefter hazards a’
Shilling to win a Groat, and the Chances be-
ing equal, it is evident there is 3 to one againft
him. : ek:
In the Second Lottery, the Gamefter ven-
tures a Shilling againit a Shilling,and the Lots
being 4 to 2, his Difadvantage is 2 tor. |
And a Lot at either of them being truly
worth juft 8 Pence, (viz. the 6th part of 3
times 16 Pence, or twice 2 Shillings) the Dif-'
advantage muft be the very fame in both Ca-’
fes, that is, the Gamefter pays a Shilling for.
a Lot that is worth but 8 Pence. e
: @.
Maifcelanea Curiofa. 67
The Method of finding this Anfwer being
mewhat out of the common Road, I fhall
re add it, and thereby infinite Solutions of
e fame kind may be difcovered. |
if. Lottery.
_ Let a=the number of Blanks
b—the number of Prizes.
v==the Value of a Prize.
2d. Lottery.
_ Let m=the number of Blanks.
n==the number of Prizes.
s=the value of a Prize.
=to what you pay for a Lot,vz. a Shilling.
$0 the Lottery has its Chances for 1, and
'Gametfter his for y—1. Now the true Odds
uifting of the compounded Proportion of
"Chances and the Values, ox. F-and >.
‘Share of the Lottery will be z, and that of
:Gamefter r4—b. Therefore as the pre-
t cafe ftands, the firft Lottery muft bea=3 —
-3b, and by the like reafoning, the fecond ':
ttery will be m=2 sx—2n”. Now the Value ©
| Lot being the Sum of the Prizes divided —
the-number-of-Lots, (which muft be equal
0th Lotteries) it yields i a
a ae
1M SH ‘
$ car ape m-\-70 %
ee: 30
68 Mifcellanea Curtofa. a |
| 14 = 3rb—3b
b 2m= 2sn—2n
| rb s2
¥ ae? | 3a +b mye
m | . | AC)
” 5\(*)
s J Cx) : :
RET NE |
= t 0. |
7* ab ee yb q
:
Scope j bai ‘
11, of alte aczo to avoid negative Numbers.
135 3 13°36 = 3qb. he Re hee
12,14 |i4qel | 7
coe _ {15'g D>1 makesa < 0,g— 1 makes 2 >
seasonal
Scope |
is 4 16 |16lfb=o
17 + 34117394 m4
16,18 N8qok
[191g <2 makes <o q > y makes b
—_—————
be fe A
397 Pntn 4
20% m-\-m1o 1 \sn = qm-|-qu
21% 2 |o9\2sn =2qm-|-2qn
24-27 ‘loslesmmm-2ng—w
22, 23. |o4\2qm-|-2qu = m-[-2n
Scope aslif m =o
24, 25 426/2q”2 = 2n
26-r2n 127951 es
as, 27 1284 >> 1makesm <0 q< 1 makesm
Ne > ~
~~ =
- Mifeellanca Curiofa. | e
A (29, 1f 2 = 0
24,29 BO2qmmm
“30 >2m Iiqmui
29, 31 ME pahegy — o Peat 0
33,4 4 (*)|34| Let vai nue Qa:
7 ee
ab 3
me a 10136) a yh = 24-|-26 = a-}- 36
: —- Blianb
20; 34. 38 Ss at 2
‘ Q * Aw, 143
3 +5 77 350 & 2m-\- 20
.3? 49. 6sn = 4m-\-4n
+ 23 *F lWti6m= 3m-6n
4°, AF
bes qm-*-4n = 3m-|6n
42— 13) ma an
; 1537 [441 = 3r—3
Baas | [+5 37 = 4
45 +2" 47 sabes
5 (*) [18]Let 4=
37, 48 [19/8 =
| 45 +3 {,0X4=4, id eft, 16 Pence.
be 6 (*) st! Let Mm 4
43> 51 52! N= 2 ;
mage 2 153 S$=2> 2 Shillings
ob
+ B 3
: Se
A New, Exatt-and Hafie Method,
of finding the Roots of any Equa-
tions Generally, and that wnt hout
any previots Reduction. By Hdm.
Halley.
a5 HE principal ufe of the Avzalytick Art,
is to bring Mathematical Problems to
Equations, and to exhibit thofe Equations in
the moft fimple Terms that can be. _ But this
Art would juftly feem in fome degree defe-
étive, and not /ufficiently Analytical, if there
were not fome Methods,by the help of which,
the Roots (be they Lines or Numbers, might
be gotten from the Equations that are found,
and fo the Problems in that refpedt be folved.
The Ancients fcarce knew any thing in thefe
Matters, beyond Ouadratick Equations. And
what they writ of the Geometrick, Construction
of folid Problems, by the help of the Parabola,
_ Ciffid, or any other Curve, were only patti-
cular things defign’d for fome particular Ca-
fes. Butas to Numerical Extradiop, there is
every where a profound Silence; fo that what-
ever we perform now in this kind, is entirely
owing to the Inventions of the Moderns.
And firft of all, that great Difcoverer and
Reftorer of the Modern Algebra, Francu Viet ay,
about 1co Years fince, fhew’d a general Me-
i
Mifcellanea Curiofa. 71
thod for extraGing the Roots of any Equati-.
on, which he publifh’d under the Title of, 4
Numerical Refolution of Powers, &c. Harriot,
Oughtred, and others,as well of our own Coun-
try, as Foreigners, ought to acknowledge
whatfoever they have written upon this Sub-
jet, as taken from Vietz. But what the Sa-
-gacity of Mr. Newton’s Genius has perform’d
in this bufinefS,we may rather conjecture (than
be fully affur’d of) from that fhort Specimen
given by Dr. Wallis in the 94th Chapter of his
Algebra. And we muft be forced to expedt it,
till his great Modefty hall yield to the Intrea-
ties of his Friends, and fuffer thofe curious
Difcoveries to fee the Light.
- Not long fince (viz. A. D. 1690) that ex-
cellent Perfon M. Fofeph Raphfon, F. R.S. pu-
blift’d his Uxiver{al Analyfis of Equations, and
illuftrated his Method by plenty of Examples;
by all which he has given Indications of a
Mathematical Genius, from which the great-
eft things may be expected.
_. By his Example, M. de Lagney an ingenious
Proteffor of Mathematicks at Paris, was en-
courag’d to attempt the fame Argument, but -
he being almoft altogether taken up in ex-
tracting the Roots of pure Powers (efpeci-
ally the Cubick) adds but little about affected
Equations, and that pretty much perplex’d
too, and not fuficiently demonftrated. Yet
he gives two very compendious Rules for the
Approximation: of a Cubical Root; one a
Rational, and the other an Irrational one.
Ex. gr. that the fide of the Cube aaa}-b, is
between
ry
eh tee Rs ee Deak tb Vide
° The an
ee
72 gto — )
Aa coal 344a- —— & Neue art --- 4 a4
And the root of the sth Power a $4 b, sh:
makes
we Viet z a pea Note, that
tis 4 aa, not 2 ie as ’tis erroneoufly Printed
in the Py ench Book) Thefe Rules were com-
municated to me. by a Friend, I having not
feen the Book ; but having by tryal found the
goodnefs of them, and admiring the Compen-
dium, I was willing to find out the Demon-
ftration- Which having done, I prefently
found that the fame Method might be accom-
modated to the Refolution of all forts of Equa-
tions) And I was the rather inclin’d to im-
_ prove thefe Rules, becaufe I faw that the
whole thing might be Explain’d ina Synopfis s
and that by this means, at every repeated:
ftep of the Calculus, the Figures already found
ya the Root, would be at leaft Trebled, which
U other ways, are encreafed butin an equal
Number with the given ones- Now, the fore-
mention’d Rules are eafily demonftrated
from the Genefis of the Cube, and the sth
Power. For, fuppofing the fide of any Cube
rate, the Cube arifing from thence, 1s asa--
344¢. £3 zee-|-eee. And confequently, if we fup-
pofe zaa the next lefs Cube, to any given Non~
cubick Number, then eee will be lefs than
Unity, and the remainder 6, will = the
other Members of the Cube, saber 3.a€e-|-e¢e.
Whence rejecting eee upon the account of its
fmallnefs, we have os ace ina And fince
| ae
Maifcellanea Curtofa. 73
aae is much greater than aee, the quantity -
will not much exceede; fo that putting
e = —- then the quantity —-——(towhich
3aa 3 4a-\-3, ae :
e is nearly equal) will be found
b b . ab
ey ih eee or ——-—— that is ——, =,
34a\-3ab 3aa+-b 344a4+-b
gaa? Dee
And fo the fide of the Cube aza--s will be
“T Saa-) :
M. de Lagney. But now, if aaa were the next
greater Cubick Number to that given, the fide
of the Cube aaa—b, will after the fame man-
ab
Fe Pan And this eafy
and expeditious Approximation to the Cubick —
Root, is only (a very fmall matter) erroneous
in point of defect, the quantity e, the remain-
der of the Root thus found, coming fomething
lefs than really ’tis.
As for the/rratiovale Formula, tis deriv’d from
the fame Principle, vz. b= 34ae--3aee, or
which is the Rational Formula of
ner be found to be 4«—
b
—_ = aelee, and fo V { aa aa é = 241-e, and
pa ae oe ag ;
/ / rs? |.ja=a+e, the Root fought. Alfo
¥ 34
the fide of the ‘Cube aaa—b, after the fume
aero _|
manner, will be found to be bat taa— b
aif M ifcellanea Curiofa.
And this Formula comes fomething nearer to
the Scope, being erroneous in point of exce/s,
as the other was in defeé, and is more accom-
modated to the ends of Pra¢tice, fince the Re-
ftitution of the Calculus, is nothing: elfe but
the continual addition or fubftraGion of the
A aee LL ) VI peer ay
Quantity according aS the quantity e can
be known. © So that we fhould rather write
2 ‘ —~€£CL ee — ; bl / a
/ bail |-ta; in the former.cafe, and in
| 34
the latter, say va pss, But by cither
of the two Formulas, the Figures already
‘known inthe Root to be extracted, are. at
teaft Tripled; which I conclude will be very
grateful to-all the Students in. Arithmetick ;
and I congratulate the Inventor upon the ac-
‘count of his Difcovery. Bt
But that the ufe-of thefe Rules may be the
better perceiv’d, I think it proper to fubjoin
an Example or two. Let it be propos’d to
find the fide of the double Cube, or aaa--b=2.
b
Here 4=1, and a = 7:&f0! -- /72,0r 1, 26,
be found to be the true fide nearly. Now, the
Cube of 1, 26, is 2,000376, and fo 0,63 +4-
\/'23969—32000376 | oF 0, 63 ty/, s9680052
ee as |
9100§291 = 1,259921049895--3 which in 13
Figures, gives the fide of the double Cube,
with very little trouble, viz. By one only di-
vifion,-and. the extraction of the fquare' Root;
when as by the common way of working, how
Dil much
A,
ak
REE Lin)
Mifcellanea Curiofa. 75)
much pains it would have coft, the Skilful —
very well know. This Calculus a Man may.
continue as far as he pleafes, by encreafing the.
Square by the addition of the quantity - ts
which Corredion, in this cafe will give, but.
the encreafe of Unity in the 14th Figure of
the Root.
. Exemp. 2d. Let it be propos’d to find the
fides of a Cube equal to that Englifh Meafure
commonly call’d a Gallon, which contains 231.
folid Ounces. The next lefs Cube-is 216,
whofe fide 6= 4, and the remainder 15 =4 ;
and fo for the firft Approximation, we have
s+-V 91-5 — the Root. And fince /9,8333...
is 3,1358..., “tis plain that 6,358 4+-e.
Now, let 6,1358=2;and we fhallthen have for
its Cube 231,0008 5 3894.712,&according tothe
Rule,3,0679'b-v/9, 41201041—, 000858394712,
OES AOTO
is moft accurately equal to the fide of the gi-
ven Gube, which within the fpace of an Hour,
I determin’d by Calculation to be 6.1357924
3966195897, whichis exactin the 18th Figure,
defeGtive in the roth. And this Formula is
defervedly preferable to the Rationale, upon
the account of the great Divifor, which is not
to be manag’d without a great deal of Labour;
whereas the extraction of the fquare Root,
proceeds much more eafily, as manifold Ex-
_. perience has taught me.
But the Rule for the Root of a pure Surfo-
lid, or the 5th Power, is of fomething a higher
Enquirv, and does much more perfectly yet,
do.
itl Shea a oh ot ABS i ma
re 3
76 |) Mifcelanea Curiofa. —
do the bufinefs; for it does at leaft Quintuple
the given Figures in the Root, neither isthe —
Calculus very large or operofe. Tho’ the Au-
thor no where fhews his method of Invention
or any Demonftration, altho’ it feems to he
very much wanting ; efpecially fince all things
are not right in the printed Book, which may
eafily deceive the Unskilful. Now the sth
power of the fide 2+e is compos’ of thete
Members, 4°-\52*e|-10@3e” -|-10a*e3-|- 5. ae*-
ei==a'-b;from whence b=sa*e |-1 oate*-}-10
a? 31-520", rejecting e* becaufe of its fmalnefs.
Whence = = a’ @-\-2ae? -|-2.4e3--e*, and ad-
ding on both fides ;2*,we fhall haveV tats ob
ant f 5a* sennodigtt “\-220! “| e*==*4a-1ae-lee.
Then fubftracting jaa from both fides, sate
¥ Re arene eg
bd Pale Se Vy 446 = ° é
will 5 hale a 9 to which if: a be
added, then will 4 e—%4 Fal pee p Petre
eoehianns a bit signed, pox
= the root of the Power 4‘+¥. But if it had
a‘—b (the quantity 2 being too great ) the
a Rule would have been thus,'4 Wa 6 --\4a.
| | 54
And this Rule approaches wonderfully, fo that
there is hardly any need of Reftitution. —
But while 1 confidered thefe things with
my felf, I light upon a General Method for
the Formulas of all Powers whatfoever, and
(which being handfome and concife enough)
- | } thoughtcI would not conceal from the Pub-
lick. : pen "Thefe*
fact bear or SNE
Maite Cu wee a7 es
4; hefe Formula’s, (as well the Rariowal,, as
the Irrational eee) are thus.
penne
| ao
Vv? a be ab Vee !a0— or 4-- oF ae ;
. | 34aa--b
OMe Vai ly == OF at
fia" “+b = 3 4-|- Vs 16 aat- — re
if Sars pee 4a+ Va A aa+ Treo
“/74 pte’ 6 aa- iy ; ——;,ora}+_47__
Ia. 7A
And fo alfo of the other a P
if 2 were affumed bigger, than
fought (which is done with fom advantage,
as often as the Power to be Refolved, is much
nearer, the Power of the zext Greater. whole
“Number, than of the zexr lefs) in this cafe, —
Mutatis Mutandis, we shall have the fame Exe
Pepons of the Roots, viz.
NS POS ee Pe ie Bly Ae RN eR, peas:
“ re 2
2B | Mifcellanea Curiof
é
a—ih
V aa Vaa-b, or 4—-——
24
/' oped, or — a
3a” Bk :
Vv‘ tbrtat Vena! =! or am a
% - ¥ pie h LAs | ab bs
4 , amy
tee! ae 104? 54° 2b
ace ey : ab
¢€ a
4/ r fra ats
V aka tal \ lr beet Mca 7 ae aj
ri ‘ ‘
; “a b
V a’—b = a-\- NV ikeaiel “hos of 4—-
214?
18; Tle ne
ogg’ within: thefe two Terms, ‘thé true
Root i is ever found, being fomething nearer
to the Irrational than the Rational Expreffion.
But the quantity e found by the frrarional For-
muja, is always too great, as the Quotient re-
fulting from the Rational Formula, is always
too little.And confequently,if we have -|-b,the
Irratioxal Formula gives the Root fomething
greater thanit fhould be,and theXatioval fome-
thing lefs. But contrary wife if it be—b.
And
Maifcelanea Curifa. 79
And thus much may fuffice to be faid, con-
cerning the extraction of the Roots of pure
Powers; which notwithftanding, for common
Ufes, may be had'much more eafily by the
help of the Logarithms. But when a Root is
to be determin’d very accurately, and the Lo-:
garithmick Tables will not reach fo far, then’
we muft neceflarily have recourfe to thefe, or’
fuch like Methods. Farther; the Invention
and Contemplation of thefe Formule, leading |
me toa certain Univerfal Rule, for adfected.
Equations (which I hope will be of ufe to all
the Students in Algebra. and Geometry) I was.
willing here to give fome account of this Dif-
covery, which I will do with all the perfpe-
cuity I can. I had given at N% 188 of the
Tranfattions, a very eafy and general conftru-
ction of all adfected Equations, not exceeding:
the Biquadratick Power; from which time
had a very great defire of of doing the fame in
_ Numbers. But quickly after, Mr. Raphfon
Meem’d in great meafure to have fatisfy’d
this Defire, till Mr. Lagvey by what he
had perform’d in his Book, intimated that the.
thing might be done more compendioufly yet.
Now, my Method is thus.
Let z the root of any Equation, be imagin’d
to be compos’d of the parts 2+ or —e, of
which, let 2 be affum’d as near z as is poffible;
which is notwithftanding not zeceffary,but only
commodious. Then from the Quantity a-+e or
a—e, let there be form’d all the Powers of z,
_ found in the Equation, and the Numerical
Co-efficients be refpectively affix’d to them :
-_ Then let the Power to be refolv’d, be fub-
ftracted; from the fum of the given Parts ( il
en et a ee a ee
GY y Es wh Saas
rc . a
80. § MifceHanea Curiofa.”
the firft Column where eis not found) which
they call the Homogeneum Comparationis, and
let the difference be +b. In the next. place,
take the fumof all the Co-efficients of e in the
fecond Column, towhich put=s. Laftly, in'
the third Column let there be put down the
fum of all the Co-efficients of ee, which fam.
calle. Then will.the Root z ftand thus in
; : : sb
the Rational Formula, viz. 2 = a-|-
and thus in the Irrational Formula, Vix. ©
sor) ORS Ae Vs ssE bt; which perhaps it
ie ¢
m ; f t ‘ ,
may *be worth while to Iluftrate by fome
Examples. And inftead of an Jnftrument, let
this Table ferve, which fhews the Genefis of
the feveral Powers of ate, and if need be,’
may eafily be continued farther, which for
its ufe I may rightly call a General Analytical.
Speculum. The forementioned Powers arifing”
from a continual Multiplication by 4--e (=z)
come out thus with their adjoyned Co-effici-—
ents. See the Table. But now, if it be a—e=z,’
the Table is compos’d of the fame Members, -
only the odd Powers of e, as ¢, ¢3, e?, e? are
? 4 9 »)
Negative, and the even Powers, aS e*, e4,e*,
Affirmative. Alfo let the fum of the Co-efhi-’
cients of the fide e, be = 5; the'fum of the’
Co-efficients of the Square ce=¢t, the fum of:
the Co-efficint of e* = u; of et = wm ,0fe=x,
of e®=y,&c. But now, fince e is fuppofed™
only a fmall part of the Root that is to be en-—
quir’d, all the Powers of e, will be much lefs
than the correfpondent Powers of a, and fo far +
the
554th?
2
MifceHanea Curiofa. 81
the firft Hypothefis; all the fuperior ones
may be rejected ; and forming a new Equati-
‘on, by fubftituting ate—z, we fhall have (as
was faid) +-b—=se | tee. The following Ex-
=
~amples will make this more clear.
Example I.
_ Let the Equation z2!—3z?-| 75 z—=10000,
be propos’d. For the firft Hypothefis, let
=10, and fo we have this Equation,
24==1a* yae tae? ae? \e*
—dz2#——da af! ae—de* ‘
Tez—=|e2 ce |
==T10c00 4000e |6o00ee 4oe? tet
— 300 G60e —~ 3¢e
Hyg FW): 738
——10000
4-450—- 40tge +597¢e—40e? -e’—=o
APs Seme s t U .
The Signs -- and — with refpe& to the
Quantities e and e’, are left as Doubtful, till
it be known whether e be Negative or Affir-
mative ; which thing creates fome difficulty,
fince that in Equations that have feveral
Roots, the Homogenea Comparationis (as they
term them) are oftentimes encreafed by the
minute quantity 2, and on the contrary, thar
‘being increafed, rhey are diminifhed. But the
Sign of eis determin’d from the Sign of the
Quantity 6. For taking away the Refolvend
from the Homogeneal formed of a2; the Sign of
se (and confequently of the prevailing Parts
in the compolition of it) will always be con-
Tt: G trary
—y
32 © ©Mifcellanea Curiofa,
trary to the Sign of the difference 6, Whence
"twill be plain, whether it muft be -| e, or —e 5
and confequently whether a be taken greater
or lefgthan the True Root. Now the quan-
tity e is ==}s— Nf Benth when 6 and ¢ have the:
t
fame Sign, but when the Signs are diffe-
Soe
rent, e is = yet But after it is
found that it will be! ~e, let the Powers e, e?,
and e', @c. in the affirmative Members of
the Equation be made Negative, and in the
Negative be made Affirmative; that is, uf
them be written with the contrary Sign. On
the other hand, if it be +e (let thofe fore-
mention’d Powers) be made Affirmative in the
Affirmative, and Negative in the Negative
Members of the Equation.
~ Nowwe have in this Example of ours,1o450
inftead of the Refolvend 10000, or b=+-
whence it’s plain, that 4 istaken greater Ran
theTruth,and confequently,that *tis—e. Hehe
. the Equation comes to be, 10450—4o1§ e -|-
§97ee—4e*-te* = 10000. That is, 450-4015 e
b5o7ce 0; and fo 450 FA4OlS €—§97 € ey
orb=se— tee, whofe Root ¢= oo V $s bt
t
or — Ag le that is in the 'prefent cafe,
at Ph 5 i 2
= ETSI 7514068 from whence we have
597
the Root fought, 9, $86, which is near the
‘Truth. But then fubivting this. for,; a, fe-
| cond
Sy
‘
Maifcellanea Curiofa. 83
cond Suppofition, there comes 2t-e=z, moft
accurately ap 8862603936495 . . fcarce ex-
‘ceeding the Truth by 2 in the laft Figure, Uz.
| bs Vases sobbe —isme. And this (if need be)
may be An much Giri verified »by Be
ing (ifit be+-e) the quantity aia
from the Root before found 5 or (it it bee )
yei—te&
by adding ot to that Root. Which
v3 45S—t
Compendium is fo much the more Valuable,
in that fometimes from the firft Suppofition
alone, but always from the fecond, a Man
may continue the Calculus (keeping the fame
Coefficients) as far as he pleafes. It may be
noted, that the fore-mentioned Equation, has
alfo a Negative Root, wz. z= 10,26.
which any one that has a mind, may deter-
min more accurately.
Example II.
Suppofe 1 1% $42 350, andletam
fo. Then according to the prefcript of the
Rule,
rt o+2= 3 34° =rs ae -|-eF
adzis Eye ee
dy speaeacat ge be
G 2 . That
= Wha ce ent eae
1 7 daira
84 Mifcellanea Curiofa.
b t
That is, -l-ro0o0t300e+30e? Fe?
+1 900—-340e771 Je”
+-540-+-§4¢
350 |
OT. 5 LO —510 T14e nat 3ee 1 cele? = 0. Now, fince
we have —510, it is plain, that 2 is aflumed
Jefs than the Truth, and confequently that e
is Affirmative. And from (the Equation )
s1O= 14e 24 Fe7 5 COMES € = V bets —Zz5
t
visa Whence z= 15,7..., which is
too much, becaufe ef z taken wide; therefore
Secondly, let a=15, and by the like way of
Reafoning, we fhall find ex 2s— (Sib
Lapeer reiaae ai
¢t ‘ i
I
= — 2G08 hati “9 and confequently <=
One
14,954068. If the Operation were to be re-
peated the third time, the Root will be found
conformable to the Truth as far as the 25th
Figure; but he that is contented with fewer,
by writing tb te? iclatita of ¢6, or fubftra-
Ging or adding ae or = sod EE the Root before
found, will prefently obtain his end: Note,
the Equation propofed, is not explicable by
any other Root, becaufe the Refolvend 350s is
greater than the @ube of = a
«Example
; he A te , af
#
MifceHanea Curiofa, 85.
Example Mil.
Let us takethe Equation z*—80 z5-1-1908
27 149372 b50c0o=0, which Dr. Wallis
ufes Cap. 62 of his Algebra, in the Refolution
of a very difficult Arithmetical Problem where
by Vieta’s Method he has obtain’d the root moft
accurately;and Mr. Raph/oz brings it alfo as an
Example of his Method, Page 25, 26.. Now
this Equation is of the form, which may have
feveral Affirmative Roots, and (which increa-
fes the difficulty) the Coefficients are very great
in refpect of the Refolvend given.
But that it may be the eafier manag’d,
let it be divided, and acccording to the
known Rules of Posting, let —z*\ 823 -
20 z? 15 z=0,5 (where the quantity zis ¥;
of z in the Equation propofed) and for the
firft Suppofition, let a= 1. Then |-’*— 5e—
2e?-|-4e?—e4— 0,5 0 5 that is, 14= ge b2ee 5
‘thence e= V3 sstbt — 3 is =37—5, and fo
x» ¥
z=1,27;, Whence ’tis manifeft that 12,7 is
near the true Root of the Equation propofed.
Now Secondly, let us fuppofe ~=12,7, and
then according to the directions of the Table
of Powers, there arifes .
b $ t u
—=26014, 4641-8193, 532e—967, 74¢%7—-50, 8e3 — EF
-\-16g870, 540-|-38709, 60e-+3048 ¢7%-|-30 @
322257, 42 —-50749, 2 e—1998 ¢?
“189699,9 -|-14937, ©
mm 5900s
7 G 3 That
86 Mifcellanea Curiofa.
That is, +298, 6559-5296 132e-+82, 26 e*
\-29,2e?—e* =03; And fo —298, 6559=—
§296, 132 ¢182, 26¢ee, whofe Root e (accord-
ing to the Rule) = zs Byte comes tQ
?
i | +3
2.648, 066—./6987686, 166022
ap D ee
| 825 2646 ee
50§644080331....ee lefs than the Truth. But
that it may be corrected, ’tis to be confider’d
14
: Fan 0000 20% we. as @ \
ne ue? --%e ) or’ 0026201 ulighedateoog;
! a/ Liss bt 26 4.354.23% AER ICE Sal's
i]
\
pee
ee / , 55--bt--tue*- te* or which is all one
3
ty a
2.648, 066--4/6987685, 67496597577 -.+-
9257126 J
5 0§644179448074402 =e; whence atemz
the Root is moft accurately 12, 75644179448
074402... as Dr. Wallis found in the fore-
mentioned Place; where it may be obferv’d,
that the repetition of the Calculus does ever —
triple the true Figures in the affumed a,
I ayer ge Te
3 ze
tg mee bt apes
quintuple; which is alfo commodioufly done
by the Logarithms. But the other Correction
t
QD
i
which the fir ft correétion,or
PA Se ey
Maifcellanea Curiofa. 87
_after the firft, does alfo double the number
‘of Figures, fo that it renders the affumed al-
together Seven-fold ; yet the firft Correction
is abundantly fufficient for Arithmetical ufes,
for the moft part.
But as to what is faid concerning the num-
ber of Places rightly taken in the Root, |
would have underftood fo, that when a is but
i part diftant from the true Root, then the
Frit Figure is rightly affumed ; if it be within
gb0 part,then the two firft Figures are rightly
affumed ; if within 7...5 and then the three
fir are fo; which confequently manag’d ac-
cording to our Rule, do prefently become
nine Figures. ,
' It remains now that I add fomething con-
cerning our Rational Formula, viz. e=- :
ss-rb
which feems expeditious enough, and is not
much Inferior to the former fince it will
triple the given. Number of Places. Now
having formed an Equation from a:te =x, as
before, it will prefently appear, whether 2 be
taken greater or lefs than the Truth ; fince
se ought always to have a Sign contrary to
the Sign of the difference of the Refolvend,
and its Homogeneal produced from 4. Then
fuppofing -\-b--se-|-2— tee =o, the Divifor is.
ss—tb, as ofren as ¢ and b have the fame Signs;
-but it is ss-\-bt, when they have different ones.
But it feems moft commodious for Practice, to
write the Theorem thus,e= ESN Keo fince
this way the thing is done by one Maultiplica-
tion and two Divifions,which otherwife would
require three Multiplications, and one Divi-
t : Z G
cette 4 aun eet
88 Maifcellanea Curtofa.
Let us take now one Example of this Me- —
thod, from the Root (of the foremention’d
Equation) 12,7..-., where
298, 6559—.§296, 132-182, 26¢ee129,
“+b is i ae
peel Sale
2e3 —e* =0, and fi nT tae ees that is, let
it\be as s tot, fo b to ah §296,1 32)2.98,
S
6§§9 into 82,26 (4,63875..- wherefore the
Divifor is s—- —. = 5291549325 ae ..) 298,
5 as
6559 (0, O§6441.-... e, that is, to five true
Figures, added to the Root that was taken.
But this Formula cannot be corrected, as the
foregoing Jrrational one was; and fo if more
Figures of the Root are defired, ’tis the beft
to make a new Suppofition, and repeat the
Calculus again: And then a new Quotient,
tripling the known Figures of the Root, will
abundantly fatisfie even the moft Scrupulous.
A Differe
Mifcellanea Curiofa. $9
A Differtation concerning the Con-
- firuction of Solid Problems, or
Equations of the third or fourth
Power, by the help of one (given)
Parabola and a Circle. iit:
: By Edmund Halley.
¥ YOw all Equations (that involve the third
or fourth Power.of the unknown Quan-
tity) may be conftructed by the help of any
given Parabola and a Circle, the Famous M.
Des Cartes has fhewn and clearly demonftrated
in the Third Book of his Geometry. But he
firft of all orders the fecond Term of the E-
' quation (if it be there) to be thrown out, and
then by the Rule there delivered, to find the
Roots of the Equation fo reduced.
‘ And fince that Operation feems too La-
borious, fome thought fit to invent a like
Conftruction, without any previous Reduttion.
_- Amongft whom Frazcis a Schooten has offer’d a
Method (for conftructing cubical Equations
-howfoever affected) which might have been
called very eafie and fimple; if (by unfold-
ing the Principle from whence he deduced his
Rule) he‘ had better confulted his Reader’s —
Memory, which he burdens with very many
and perplex’d Cautions. But lately our Fa-
mous Countriman, Mr. Thomas Baker, in a
whole Treatife written upon thefe Conftru-
3 oe | | étions,
go me Mifcellanea Cuntafa. co!
tions, has comprehended not.only all Cubi-
_ cal, but alfo Biquadratical Equations of every
kind, under one General Rule, which he has
~ demonftrated, and abundantly Iiluftrated with
Examples through ail Cafes; and moreover
at the Clofe, propos’d a way, by which that
Gencral Rule might be Invefligated. But he
does not fhew the very Method, by. the-help
of which (as I fufpect) he obtain’d his Umi-
ver {al Geometrical Clavis, or at leaft might have
obtain’d it with much more eafe. And fince
this Rule of Baker’s is no lefs perplex’d with
Cautions about the Signs +> and — than Schoo-
ten’s is, fo that a Perfon can hardly perform
thofe Conftructions aright, without he has the
Book by him; I thought that it wou’d not be
either Unpleafant or Unprofitable to young
Students, to. explain the Foundations of both
Rules, and by fome emendation of the Me- |
thod once more, to afford as much light as I
coud in fo dificult a Matter. Cartefius's Con-
ftruction (which does very eafily difcover the
Roots of all Cubick or Biquadratick Equati-
- ons, where the fecond Term is wanting) may
be fuppos’d as known. Yet fince ’tis the main
bottom, on which all that follows does de-
pend ; that this Diflertation may not feem to
want a principal Part, I'll here add the Rule
taken out of his Geometry, altering fome few
things (as I think) for the better.
- The fecond Term being out of the Equati-
on; all cubical Equations, are reduced to |
this Form, z’y. apz. aaqeo3 and Biquadra- -
tical ones to this Form, <*. y. apzz. aaqz.
a*r =o, where a.denotes the Latus Rectum of
any given Parabola, which, is ufed ~ the
one
Mifcellanea Curiofa. 91
ConftruGtion. Or elfe- taking @ for Unity,
thofe Equations are reduced to thefe Forms,
viz. Cub. z3. yp x. g =0,and Biquadr. z*y.pzz,
qz.r.=t 0. Now the Parabola FAG, Fig. 9 be-
ing given, whofe Axis is ACDKI, and Para~
meter= zort; let AC be taken= 2 4, and
be fet off always from the Vertex _4, towards
_ the inner parts of the Figure. Then take CD
2 p, in that Line AC, continued towards C, if
it be ---p in the Equation, or towards the
contrary Point, if it be tp. Farther,from the
point D (or from the point C, if the quantity
_ pbe not in the Equation) Let DE (erected
perpendicular to the Axis) be made= 2g
which is to be fet to the right hand if it be
---g, but to the other fide of the Axis if it be
--g. And then a Circle defcribed on the
Center E, which the Radius AE (if the Equa-
tion be but a Cubical one)-will interfe@ the
Parabola in as, many Points (viz. F, G, G,) as
the Equation has True Roots, of which the
Affirmative ones, as GK, fhall on the right
fide of the Axis, and the Negative ones as
FL, on the Left. But if the Equation be a
Biquadratical one, then the Radius of the Cir-
cle AB, by adding (if it be ---7) or Subftraa-
ing (if it be -v) from the Square of it, the
Re@-angle 4 «7, or the content under the. Pa-
rameter, and the given Quantityr; which is.
very eafily done Geometrically. And the In-
terfections of this Circle with the Parabola,
will give (letting fall Perpendiculars from
thence tothe Axis) all the trve Roots of
the Biquadratical Equation; the Afirma.
- tive ones being on the Right fide of the Axis,
‘and the Negative ones, on the Left. The de.
ie eS arcs demon-
92 Mifcelanea Curiofa.
monftration of all which I leave to Cartefius
‘the Inventor. Let it be Noted, that I endea-
vour here that the Affirmative Roots, may
always be had on the Right fide of the Axis,
to avoid the Confufion that will neceflarily
arife from a multitude of Cautions, where the
reafon of them is not evident.
Having premifed thefe things, in order to
make way for the conftruction of thefe Equa-
tions, even when the fecond Term is found
in them, we are to confider the Rule it felf
for taking away the fecond Term, and redu-
cing the Equation to another, fuch as might
be conftruéted by the foregoing Method. Now
all Cubick Equations of this Claffis, are re-
duced to this form, z* 6zz. apz. aaq =o, or to
this, 2%. bz”. y. aaqgz=o. Biquadratick onés
may be reduc’d to this, 2*. bz. apz*. aaqz.
a’r=o, or this, z*. or this z*. 623.4. aagz.
areio, or this, z*. bz3. apz?. x. air mio, Or
laftly, to this Form, z*.bz5 y. y. @r=o,
From all which there arifes a great Variety,
according as the Signs + or — are diverfly
connected, together ; and hence the General
Rule ferving all thefe cafes, is rendred very
obfcure and difficult, unlefs (manag’d by the
help of the following Method) it be cleared
up and delivered from thofe Intricacies. _
The fecond Term in Biquadratical Equati-
ons, is taken away by putting «=zT Jb, if it ~
be -+d in the Equation; or «=z—7), if it be
—b. Hence x; 6 in the firft cafe, and x.
+b in the fecond, is = 3 and fo in any Equa-~
tion propofed, fubftituting inftead of z, its
Equal, there will come forth a new Equation, |
wanting the fecond Term, all whofe ik
— Mifcellanea Currofa. 92
do exceed, or come fhort of the fought’ Root
z, by the given difference 44. But fince in
things of this kind, Examples do more than
Precepts, let us propofe one or two Equati-
- Ons to be conftructed.
E xample Is
uA+-b23 —aprz---aagqz aire.
put x---56=7, and then will
Mame be H- bb 2.
HF want 7b + 2.xb* --- 2,55 = 23. and
mn x3-+ 2 O x2~0- 2b) x 1,264 =. 2*
Hence it follows, that
xt eb 2b2x2 --- 3 bix + i4.b4 ~*,
bx3---3.b? x? -| 3 b3 x--- Ab = 1bz3
=--apx? |S apbec---shapb” rm ---apz”
en-au ia’ qb —a*qz
Tair
The Sum of all thefe is a new Equation
wanting the fecond Term, and which confe-
quently may be conftructed by Cartes’s Rule,
by taking inftead of 5 p, half the Coefficient
of the third Term, divided by 4 or the Para-
ae b é
meter, that is—,3 2 2p; and inftead of 3g,
half the Coefficient of the fourth Term, divi-
ia : Aah ee . pb
ded by aa, that is, Fas a i ae The
Mem-
94 Mifcellanea Curiofa. |
Members of which that have the Sign -|- are
to be fet off to the left Hand from the Axis,
and thofe that have the Sign to the Right 5
in order to find the Center of the Circle re- ©
quired for Conftruction, whofe Interfections
with the Parabola (letting fall perpendiculars
to the Axis) may give all the true Roots ~,
namely, the Affirmative ones on the Right
fide of the Axis, and the Negative ones on
the Left. But now, whenwx— b= 2%, then
drawing a Line Parallel to the Axis on the
right fide of it, and at the diftance of =), the
Perpendicularg terminated on this Parallel,
will denote-all the enquired Roots z, the Af-
firmative ones on the right fide, and the Ne-
gative ones on the Left. As for what relates
to the Radius of the Circle, it is had, by ad-
ding the Negative, or taking away the Af-
firmative parts of the fifth Term divided by
aa, from the Square of the Line AE, drawn
from the Center E found, to_4 the Vortex of
the Parabola; which is moftly done, by ta-
king iuftead of AE, the Line EO which is —
terminated at O the Interfe€tion of the Para-
bola, and the fore-mentioned Parallel; for the
Square of this comprehends all the parts of
. the fifth Term, brought into the new Equa-
tion upon the cafting out of the fecond Term,
.asis eafily proved: And it remains only, that
the square of EO be increafed, if it be. ---7,
in the Equation, or diminifh’d, if it be “Ir, by
the addition or fubftraction of the Rettangle
ar from whence the Radius of the Circle de-
fired, is compos’d. This Method of inveftiga=
ting M. Baker’s central Rule, is eafie and free
#romvall Cautions; and the difference arifes oe
y
ee wee eT oie
r4 8 ‘
7
Mifcellanea Curiofa. 95
ly from hence,that J determin the center of the
Circle, by the Axis, and he by a Parallel to the
Axis, and that 1 always have four Affirmative
Roots on the right fide the Axis, which hehas fome=
times on the right fide, and fometimes on the left.
' As for cubical Equations, they are to be
reduc’d to Biquadratical ones, before they can
be conftructed by the fame General Rule;
which is done by multiplying the Equation
proposd by its Root z; whence arifes a Bi-
quadratick Equation, in which the laft Term
or r, is Wanting. Wherefore taking away the
fecond Term, and finding the Center E, the
line EO isthe Radius of the Circle; viz,
When 47 is =o, and the whole fifth Term
in the new Equation, arifes from the taking
away ofthe fecond Term. Let this Equation
be propos‘ to be conftructed.
Example if.
Z3 = bz* -bapz Faaq= o,
which multiply’d into z, becomes
OR CAPS | BAGS EO.
To take away the fecond Term, put
Kbeg OS z, and then will
xt Chek 7 4 bb? bin 1552 b* me +4
— bx? 4 baxter Bb J bem bs
reaps? 1 2 abpx|\1,apb? eat “apes
‘bh aaqxe\)aagb = Vaagqz.
"Now in this new Equation, the half Coef-
ficient (of the third Term) divided by 4, wz.
my a |-2 p, is to be ufed inftead of z P} and
Le by adiad “at 2 bale eal, Mahi
96 Mifcellanea Curiofa.
the half Coefficient of the (fourth Tertn) di-
vided by a a, the Square of the Latus Rettum,
| b
4
pe 29, is inftead of 34 in
Vi%Z. i
16a*
‘Cartefius’s Conftruction, from whence the Cen-
ter Eis determind. Then drawing a Paral-
lel to the Axis, at the diftance £4, to the left
fide (becaufe of y= « 1-15) whofe Interfecti-.
on with the Parabola, let be O; a Circle de-
fcribed on the Center E with the Radius EO,
will cut or touch the Parabola in as many
Points as the Equation has true Roots, which
Roots,or z,are the Perpendiculars let fall from
thofe Points upon the Parallel to the Axis,
the Affirmative ones to the Right fide, and
the Negatives to the Left. Ifthe third or
fourth Term,or both, be wanting in the Equa-
tion, there’s no difference at all ( of the Me-
. thod of inveftigating the Central Rule) to be
obferv’d. But the Quantity p or g being want-
ing, thofe parts.of the Lines CD and DE (in
fome manner deduced from that Quantity
will be wanting too, and we are to procee
with the other Coefficients of the third and
fourth Term in the new Equation, according —
to the way prefcrib’d in the foregoing Exam-
- ples.
Hitherto we have confider’d Mr. Baker’s
General Method, than which none more Eafie
and Expeditious ts to be expected, ufing either
a Parabola, or any other Curve for a Conftru-
ction, viz. when the Equation rifes to the Bi-
guadratick Power. For while I am writing
of this, ’tis my good Luck to hit upon a cer-
tain Geometrick, Effection of the central Rule,
which is Expeditious beyond Hope, and will
abun- —
EE
; 2) Curiofa. 97
‘abundantly fatisfy thofe that are curidus in
thefe Matters. |
(Fig. 10.). Having defcrib’d the Parabola
NAM, whofe Vertex is 4, Axis ABC, and Pa-
rameter 4; let the Fquation be reduced to this
Form, zt. bz}. apz*.. aaqz. a?r. == 03 or
if it be only a Cubical one, to this, z}. gen
‘@pz. aaq. = o. Then at the diftance BD 4),
Jet DH be drawn parallel to the Axis (to the
Left Hand if it be—4, and. to the Right, if
it bed) meeting the ‘Parabola in the point
D, from whence let fall BD. perpendicular
to the Axis. In the Line AB continued to-
wards B, make B K==;4, and draw the Line
DK. interminate on either fide: < Harther,
take K C==2 AB, always in the Axis produ-
ced beyond K; and if the quantity p has
the ‘Sign =i, take towards the fame parts,
CE", but towards the contrary part, if
it be tp. Then at the point E (but at the
point C if the quantity be wanting) erect
EF. perpendicular to the Axis, meeting Cif .
need be) the-Line DK prodtced, in "the
point F, which point is the Center of the
Circle required, if the quantity g be want-
ing. But if g be in the Equation, then we
muft take in the Line F E (if need be) pro- -
duced the length of F G==34, which place
to the Left Hand if it beg, but to the
Right if it be —q; and then the point G
wil ll be the Center of the Circle required for
the Conftruétion, and the Radius of it, will
be the Line GD, if the quantity + be want-
ing, that is, if the Equation be only a Cubi-
‘al One; the Square of which fame Line (in
Bididdiatick pingsens) is to be i
¥
Harare nme PRTC ORCEN Se er
98 Mifcellanea Curiofa.
by the addition of the Rectangle under +
and the Latus Rectum, if it be —r, or to
be diminifhed by the fame Rectangle if it be
-+-r. The Circle thus deferib’d, and Perpen-
diculars let fall from its InterfeGtions with
the Parabola, to the Line DH, thofe that
are at the Left Hand as NO will always be
the Negative Roots of the Equation, and
thofe at the Right, the Afhrmative.
Cubick Equations are otherwife (and fome-
thing more fimply) conftruéted according to
Schooten’s Rule, m which alfo the Roots re-
fpe&t the Axis. But becaufe the Inventor
himfelf does neither explain the Inveftiga-
tion nor Demonttration, it will not be amifs
to fhew the Foundation of it here, and at
the fame time render the Geometrick Con-
itruion more Elegant, and rid it of thofe
Cautions in which ’tis involw’d.
This Rule is deriv’d from hence, that
every Cubick Equation may be reduced to
a Biquadratick one, in which the fecond
‘Term is wanting. Which is done, by mul-
tiplying the Equation propofed into z —bme,
if it be 1b in the Equation, or into z--b=0,
if it be -b; and the new Equation thus
form’d will have the fame Roots with the
Cubical one, and moreover another Equal to
+b, if it be —b in the Equation, or con-
trariwife. ©
Let the Equation z<? +2’ b--apz ---aag=0,
be propofed to be conftruéted ; multiply this.
inte <-+b, and it makes.
x mt 23D-\-apz*-|-aaqZ,
4-23 b—bbz’-|-abpz--aagb .
| Here now the fecond Term is wanting, Ai
the
+ WMifceHanea Curifa. — 99
the Coefficient of the third Term — bb-\-ap,
gives os id 4-2 p, in the room of 3p or CD
in Cartefiuss Conftruction; and from half
the Coefficient of the fourth Term is made —
&g-- iB, inftead of 3q or DE, and fo the .
Center of the Circle fought is determin’d.
Alfo becaufe one of the Roots of the new
Equation, vz. -\-b is given, a point in the
Circumference will’ be given too, and confe-
quently the Radius. | Laftly, Having de-.
icrib’d the Circle, Perpendiculars let fall
from its Interfections with the Parabola,- to
the Axis, will give the Roots of the Equa-
tion, both Affirmative and Negative, in the
Jame manner as before. :
_ Now the Center of the Circle is found by
a moft eafy Conftrudction, and which is to be
preferr’d to all others, in Cubick Equati-
ons.
_ Fig. 11. Let 4 be the Vertex, arid AF the
Axis. of the deferib’d Parabola AMD; at a
diftance equal to 6 let DK be drawn parallel —
to the Axis, to the Right Hand if it be-|-6
in the Equation, and to the Left, if it be
—b:; which Line fuppofe to meet the Para-
bola in the point D. Upon the Centers D
and A, and with equal Radij, defcribe on
both fides two occult Arches, interfecting
One-another, and thro’ thofe points of In-
terfeGiion draw the interminate Line BC
which cuts the imaginary Line AD in the
middle and at right Angles, and meets the
| H 2 : Axis
-
A eee eh Gy i i'w
mS BPs bin, DECC, Ue TORT aes =
PAR i ha Ae rer y sui
1900) Mafcellanea Curiofa.
‘Axis in E. From £ fet off EF = 3 p, down-
wards, if it be —p in the Equation, but up-
_ wards towards 4 if it be +p, then at the
| point F(or £, if p be wanting) erect the Per-
-pendicular F G, meeting the Line BC in G,
and in GF produced take GH = 34, to the
Right Hand, if it be ~q in the Equation,
but to the: Left, if--g.. Then will the point
#1 be the Center, and HD the Radius of the
Circle fought, which (¢letting fall Perpendi-
culars to the Axis from its Interfections with
the Parabola) will fhew all the Roots (as
LM) of the Equation. And how this Con-
ftrudion follows from what went before, is
evident enough of it felf, fo that there is no
need of infifting any farther upon the De-
monftration of it. | :
!
A Dif
“
4
= ‘
aay
‘
ope) ‘
~ ‘ Uy
x
. ‘ zs +e ’
. My “ ts
py) ae ~
ae ae
BN aes oot See = = =
& “S mae
é ato Noe So»
: z > Fig d .
at x
ee \
4 ’
‘
‘ ‘
a N
‘
’
'
'
. ‘
F i
‘
‘
‘
,
4
4
‘
4
4
; 2 iz
se .
= =
fe
jal aa
eae ee
Mifcelanea Curiofa. 101
LY
i 4
A Difcourfe concerning the Num-
ber of Roots, i Solid and Bi-
quadratical Equations, as alfo of
the Limits of them. :
| | By E. Halley.
| HY in the precedent Difcourfe fhewn
a Method, by which folid Problems
however affected, might be conftruéted after
‘a moft fimple and eafy manner, by the help
_of one given Parabola and a Circle ; towards
the latter end a certain pleafant Speculation
‘offer’d it felf, namely, that from thefe Con-
_ftrudctions, the Number of Roots in any E-
quation, with their Limits and Sines, would
eafily follow and be determin’d. Upon which
‘account, I promis’d that I would quickly
‘write a fhort Differtation concerning this
Subject, in which I was perfwaded I fhould
perform fomething not unprofitable nor un-
‘grateful (if not to the Geometers of the
firft, yet at leaft) to thofe of the fecond
Rank. sence He Bi
. But coming to look nearer into the Bufi-
nefs, I found I was imprudently fallen in
among fome of the profound Difficulties of
Geometry, and deftin’d to handle the fame
Things, that formerly .employ’d the Pains
of two Tluftrious eet apate and Cartes 5
‘Gabi colonel 3 il
102 Maufcellanea Curiofa.
in which they either of them (by a like Fate
tho’ in a different way) committed a Para-
logifm, perhaps the only one in all their
Geometrical Writings ; as fhall be afterwards
prov’d. Wherefore being fenfible, as well of
the Difficulty, as the Excellency of the Sub-
je&t, I refolv’d to apply to it ftrenuoufly,
that I might not be thought unable to per-
form my Promifes, and that fo noble part of
Geometry, and fo little cultivated, might
not lie any longer wrapt up in Darknefs, but
‘be render’d plain and intelligible by thefe
few Lucubrations of mine. But firft the
Reader muft take notice, that while he fets
“to the. Reading of this, he ought to have
“the foremention’d Differtation (Ao. 188.) at
‘hand by. him, and to underftand the Con-
ftructions there delivered very well; becaufe
thofe things that follow do chiefly depend
upon them, neither are they to be here re-
“peated again. oan) |
It is plain from Cartefivs, and what was
there faid, that both in Cubick and Biqua-
dratick Equations, the Roots may be expoun-
ded by Perpendiculars let fall, upon the-Axis
or. given Diameter. of the given Parabola,
‘from the Interfections of that Curve with a
Circle. And. whereas when. a Circle inter-
teas a_Parabola, it muftneceflarily, do fo,
“either in four or in two points; it’s manifeft,
that in. Biguadraticks there, muttalways be,
- Wegatives as alfo if the Circle happens to
touch it, in which cafe the equality of two
Roots.of the /ame Sign, is concluded, , But
in Gubick, Equations, becaufe,pne,of the In-
¢ Te ae terfections
Mifcellanea Curiofa. 103
‘terfections is requir’d to the Conftrudtion,
therefore either but one, or the three re-
maining Roots, do denote one or three; as
in the cafe of Contact; whence its plain,
that there are found two equal Roots, and
that the Problem from whence the Equation
refults, is realy Plaii.
Therefore all Cubick Equations however af-
fected, are explicable by one, or by three
Roots, which are always poffible, that is, if
we admit Negative Roots for trve ones. So
Biquadraticks whofe laft Term 7 is affected
with the Sign —, are explicable by two or
four ; but if it be +-r in the Equation, and
it be fo great that VGD q — ar (See Fig..10.)
be lefs than that the Circle defcrib’d with
- that Radius and on the Center G, can touch
the Parabole in any point; the given Equa-
tion is altogether impoflible, nor is it expli-
cable by any Affirmative or Negative Root;
but more of this in the following Pages.
Now fince there js fo great a difference be-
tween the Cafes of Cubick and Biquadratick
Equations, that they cannot be comprehend-
ed together, we will firft of all handle the
-Cubicks, and then the others. ‘The Cubicks
are conftruéted by an infinite Number of
Circles in a given Parabola; but the Biqua-
draticks by one alone (at leaft by thefe
_ Methods) and that becaufe, putting z---e (or
-any Indeterminate) equal to. nothing, the
Cubick Equation is reduced to a Biqudratick
having the fame Roots with the Cubick, and
befides that, another Root equal to ¢; whence
it comes to pafs that the Cubick Equation
4 H 4. may
mer aii 9 Seta Lk |
i04 Maifcellanea Curiofa.
may be conftructed by as many different Cir-
cles, as you can imagine Quantities e, that
is, an infinite Number. But among all thefe,
_ that which I gave before, is the eafieft: Yet
there is another not much inferior to this,
which feems better accommodated to the de-’
figns of determining the Number of the
Roots, and their Limits; and which arifes
from the taking away of the fecond Term,
by putting after the common way «=z
-or — } of the Coefficient of the fecond
Term. Now this way is thus: The Parabola
ABY (Fig. 12.) being given, whofe Vertex. is
A, its Axis AB, and Larus Refum a, \et the
Equation be reduced to the ufual Form, vz.
z3. bz. apz. aaq. to. ‘Then at the diftance
of ; 4 let there be drawn BK (parallel to
the Axis, to the Right Hand if it -- 4, other-
wife to the Left) which meets the Parabola
in B; and let the Line DP interminate on
both fides, be ereéted perpendicular to the
fuppos’d Line AB, meeting the Axis in the
point G. From the point B, let fall the
Perpendicular BC, and let GE be always made
equal to AC, and be fet off towards the
lower parts. From E fet off EH = 2 p, up-
wards if it be ++ p in the Equation, but
downwards if = p; and from the point A,
(or E, if the quantity p be wanting) let the
Perpendicular HQ be drawn out, méeting
the interminate Line DP in O. Laftly, in
the interminate Line HQ, make OR = ¢ 4,
from O to the Right Hand, if it:be’— 4, but
to the Left, if--g. Then a Circle defcrib’d
on the Center R with the Radius RA, will
cut the Parapola in as many points, as. the
pes Tee © ~ Equation
4
— Maifcellanea Curiofa. 105
Equation propos’d has Roots, and they will
be the Perpendiculars ZY, let fall from the
Interfections Y, to the Line BK parallel to |
the Axis; of which thofe that are to the
Right Hand of the Line BK, are the Affr-
‘mative ones, and thofe to the Left, the Ne-
gative. : Bee
_ . The conveniency of this Conftruction, lies
in this, that ’tis perform’d by a Circle paf-
fing thro’ the Vertex, in the fame manner as
if the fecond Term had been wanting. And
therefore to determine the Number of the
Roots, ’tis fufficient to know the Properties
of the Place, or that Curve Line which di-
ftinguifhes the Spaces, in which if the Cen- ©
ter of the Circle (that pafles thro’ the Ver-
tex of the Parabola) be placed, the Circum-
fetéhce of it fhall interfeét the Parabola ei-
ther in one or in three other points: That is,
to define the Nature of that Curve, in which,
fall the Centers of all the Circles pafling
_ thro’ the Vertex, and then touching the Pa-
-rabola. Now this Locus, is that very Para-
boloid, which the celebrated Dr. Wailis calls
the Semzcubical, in which the Cubes of the
Ordinates are as the Squares of the corre-
fpondent Abfcifles, The Latus. Rettum of
which, is #3 of the Latus Rettum of the gi-
yen Parabola, and its Vertex the point U
(Fig. 12.) the Line AU being half the Latus
_Rettum of the fame Parabola. That is, if
we put unity for the Latus Reétum of the gi-
ven: Parabola, then 3; of the Cube of the
ordinate applicate, will = the Square of the
‘intercepted Diameter; or the Cube of 3
YH =the Square of AR, wiz. if R be the
dil Center
106 Mafcellanea Curiofa.
Center of the Circle that pafles thro’ the
Vertex of the Parabola, and touches the fame
afterwards. : if
_ This is that Curve which our Country-
man Mr. Ne (the firft of all Mortals) de-
-monftrated to be equal toa given right Line,
and by that means obtain’d a Reputation
among the principal Geometricians. Its pro-
perties have been curioufly enquired into, by
Dr. Wallis, (at the end of his Book of the
Ciffoid ) and Flugenius (Prop. 8 & g. of his
Tra&t of the Evolution of Curve Lines) and
others, whofe Writings the Reader may con-
fult. This Curve defcrib’d on either fide of
_-the Axis of the Parabola (viz. VNL, YPX)
-comprehends a Space, in which if the Cen-
ter of the Circle (which pafles thro’ the Ver-
tex A) be placed, it will cut the Parabola in
thnee other points. But the Spaces more re-
mote from the Axis, do afford Centers for
Circles that will cut the Parabola but in one
-point befides the Vertex. e
‘Thefe things well underftood, we are now
prepar’d to determine the Number of the
Roots. And firft of all, let the fecond Term
be wanting, and let the Latus Rettum = 1, or
AV =. Inthe Conftrudion VH is= 2p, HR >
-r3q3 and fince if it be-+-p in the Equation, ©
zp is to be fet off from U towards the upper
parts, the Center of the Circle is always
found without the Space LVX, and therefore
is explicable by one Root only, which is Af-
firmative if it-he 49, Negative if --9 5 and
thefe Roots’ may be inveftigated by Cardan’s
Rules. But-if it be — p, then UH= 4p, is
fet off towards the lower parts; and it is
a3 poffible .
F) RET MAR OS SET sy eae Sm)
ee 4 i
— Mifceilanea Curwfa. 107
poflible that HR may fall between the Axis and
the Curve UX or UL, wiz. if the Cube of 2
UH or of p, be greater than the Square of 3 93
that is, if z-p* be greater than 39°; in which
‘cafe there are three Roots, two Negative, if
it be — g, and one Affirmative equal to the
fum of the others; but if it be +4, then
there are two Affirmative ones, and one
Negative. But if 2>p? be lefS than 3 97,
_ then there is but one Root, Affirmative if it
_ be 4, Negative, if -|-g. Allwhich things
are taught by thofe that have handled this
part of Geometry. |
Now let all the Terms be in, and firft let
there be propofed, as an Example, this Equa-
tion, z* — 2” b\-zp-q =, to which Fig.12.
ferves. In-the Conftruction of this, we have
BCH:b, UGas AC 3’, VES yO’,
UH=26* 4317p, GH=36 tp, orip |
36°. Hence HO= i, b? 4 45, or Shp me
27 6*, and HR (that is the diftance of the Cen-
ter of the Circle R from the Axis) is ever the
difference between 3 bp and ,', b* +2 4,which
Expreflions if they are equal, then the Center
falls in the Axis: If 2 bp be greater than ;, b°
+49, then it falls.to the Left Hand of the
Axis, if lefs, then to the Right. If therefore
the {quare Root of the Cube of 4 UH (that is
of 3 6? 1p, or putting 5 67— ppd,
if /ddd) be greater than HR, that is the
difference between ,7 0' 39 and 2 bp; the
- Genter R will be found within the Space —
NPU circumfcrib’d by the Paraboloids UPX,
UNL, and the interminate right Line DNP;
and fo the Circle will cut the Parabola in
three points Y, Y, Y, pofited to the Right
Hand
BA Ai
108 = Mie sane’ Curiofa
Hand of the Line BK, and fo the Equation will
have three Roots. But the Center being with-
out this Space NUP, it is explicable but by
one Affirmative Root. ‘Here it may be noted —
by the by, that the Right Line DP may touch
the Paraboloid UPX in the point P, EP be- —
ing #, 6? ; but will cut the other Paraboloid
UNL in the point NV, fo that letting fall NF
perpendicular to the Axis, UR ist \ EUs or -
3407, and NF yo¢ 0%. But UW (which be-
ing perpendicularly applied to the: Axis at
the point UV; meets DP-in W) is = 4 b>, or -
7 EP?
* Hence we may 4 fafely conclude, that if-in
the Equation either p be greater than $67, _
or gq greater than ,, 6%, that there will he |
found but one Root, and that an Affirmative
one. Carte’s Rule therefore (Page 70. Edit.
Amftera. 1659.) is net true, in which he de-
termines that there are always as many true
Roots, as there are changes of the Sines +-
and — in the Equation: Schooten in his Com-
mentaries vainly endeavouring the defence
‘of this’ Miftake. Alfo Prop. 5. Sect 5 OF
our Country-man Aarriot’s Ars Analytice (as
alfo Prob. 18. of Vieta’s Numer. Poteft. Retol.)
is hardly found; fince from the Limitations
- which they have there fet down, that muft
agree to the whole Parallelogram PIUW,.
which we ‘have prov’d does agree .only to-
the Space NUP; that is to afford a Center
to the Circle interfecting the Parabola in
three other points befides the Vertex.
But the quantity q or the laft: Term @
and P peut Sar fo that ? be pers
—Mifeellanea Curwfa. 0g
3 6?) is exadly limitted from the foregoing
Equation Vddd= 3b +iqné bp; viz
when the Circle touches the Parabola. There-
fore; q ought to be lefS than ¢4p 37,07
Va} ; but if p be greater than 407, alfo; 4
ought to be bigger than ¢ bp 4 27 6? a a/d?,
that the Center may not fall in the little
Space NUW. And with thefe Conditions
the Equation will always be explicable by
three Roots; otherwife but by one. But
whether there be three or one, they are al-
ways Affirmative ones, becaufe of the pofi-
-tion of the Center R to the Right Hand of
the Line DP. | ie
_ And this is the moft difficult Cafe; fo that
thofe that well underftand what has gone be-
fore, will without any trouble take what
comes after. Now let the-Equation 2? —é |
xz? pz qo, be given. Here (that there
may be three Roots had) the Center of the
Circle ought to be found fomewhere within
the Space PNa, determin’d by the right
Lines PN, Pa, and the Curve of the Para-
bola Na; wherefore fince EF is= 3 00, p
ought to be lefs than 4 4b. Now for the de-
termination of the quantity 7, ¢ being = %
b? 3p as before, /d? +27 6? — % bp ought
always to be greater than} 9, that fo the
€enter of the Circle may be pofited in the
forementioned Space PNa; which when ’tis
fo, fuch an Equation has two Affirmative
Roots, and one Negative. But if p be greater
than } bb, or 39 greater than /d’ -l- 27% =
é op; it is explicable but by one (and that a
Negative) Root, Let
Let
FIO Maifcellanea Curtofa.
Let the Equation 23 mbz? —pz—qma, be
propofed in the next place. That this Equati-
on may have three Roots, the Center of thé
Circle muft be found fomewhere in the inde-
finite Space between the right Line DPD
and the Curve of the Paraboloid PX. The
quantity p is not here liable to Limitations ;
but 3 7 ought always to be lefS than /a? —
z7 6? — 4 bp, fappofing d to be =% 6? + 3 p.
By this means, there are two Negative
Roots afforded, and one Affirmative ; but
otherwife, if ¢ 7 be greater than Wd? — 5
6? — & bp, the Equation is explicable by one
only (Affirmative) Root.
Fourthly, Let the Equation 27 — bz — pz
‘\+q=0, be propofed, which has two. Affir-
inative Roots, and one Negative, if the Cen-
ter of the Circle be found in the indefis
nite Space between the right Lines Pa, PD;
- and the Curve of the Paraboloid ak that
is, Cputting d= 5 bb Fp) aq ve lee
than 4/d* + 35 b? -\- bp; but if 7 9 be’ great=
er than this quantity, there is but one (Nes
gative) Root. ee i
But the four remaining Equations in which
we have 1b; do not differ from thofe that
have been mention’d: already, as to the Limi-
tation of the Number of the Roots, if the Sign
of the laft Term be changed, keeping the
Sign of the third Term. But then them that
were the Affirmative Roots-in the former, —
will be the Negative ones here, and comtra- ~
riwile. Wa)
Thus in the Equation <? —-bz* -pzmq -
xt e, the Affirmative Roots. were either one
or three; but in this Equation 2? be? pz
ASE alti dae apn’ a
“By
— Mifeellanea Curiofa. 111
‘\- q = 0, there is either one or three Nega-
tive Roots, under the very fame Conditions;
but no Affirmative Root at all. So alfo in
the Equation 2) -\-bz?-\pz—qs0, there;
are two Negatives and one Affirmative, if p-
be lefs than 3 6b, and 3 lefs than /43 +3
~ b? — & bps even as in the Equation 2? — bz?
th pz qo, there were two Affirmatives
and one Negative: But the quantities p and
q exceeding thofe prefcrib’d Meafures, there
is here only one Affirmative Root, which there
was a Negative one. In like manner, is the
Equation 2? bz? -pz-+-q=o0, there are
either two Afirrmatives and one Negative, or
one Negative only |
Laftly, For the fame reafons in the Equa-
_tion 2° bz? 4—pz—q=o0, there are two
Negatives and one Affirmative, or one Af-
firmative only, for which, in the Equation
Re mt bx? — pz lg = 0, there were two
Affirmatives and one Negative, or one Ne-
gative alone; vz. aS 34 is either greater
or lefs than /d* -\- ,> b> -\-@ bp.
If the third Term (or pz) be wanting, the
Center AK always falls in the Line IPEa,
wherefore if it be <3? — bz”. y. 4 9 or 2? 4
ba. x. tg, there can. be byt one Root,
Affirmative if it be - 6, Negative, if -\ 2.
But if it be 2? bz". y. + ¢ or-z3 + bz?.
x. 9g, there may be two Aifirmatives and
one Negative in the former, or one Affiir- -
mative and two Negatives in the latter, the
Center falling in the Line Pa between P and
A, that is it.4.q be lefts, than 2707.5 for if
it he greater, there can be but one Negative .
in the former, or one Affirmative in the lat-
ie an i Hitherto
132 MzfceNanew Curiofa.
_ Hitherto we have obtain’d the Number of
the Roots in €ubick Equations, it remains
that we add fomewhat concerning the quan<
tity of the Roots. And here it is firft of all
to be noted, that every Equation having three
Roots, may be expeditioufly enough refolv’d
by the help of the Table of Sines, that is by-
_ the Trifection of an Angle, by- putting
a/ 4b? — +p or Vad = the Radius of the Cir- |
cle, if it be -- p in’ the Equation; or
V£b7 +4, if — p; and the Angle to he
Trifeéted, that which has its Sine (in the ~
Table of Sines) 2,0? ¢ bp tag. This —
Angle being found, the Sine of its third.
part, as alfo the Sine of the third part of
its Complement to a Semi-circle; and their
Sum, will be given from the Table of Sines.
Now thefe Sines are to be multiplied into
the Radius V 4 67-4 p, and thus will be ob-
tain’d the quantities (yv& y& y& in the iz.) the
Sum or Difference of which and + 6, as the
cafe requifes, will give the true Roots of
the Equation. All thefe things are deduced ©
from Cartes’s Difcoveries. But that I may —
comprehend all the Cafes, with 4s much Bre- ©
vity as is poffible ; I fay, that the Center R,
in the firft Formula of Equations, falling in
the Space UGP, the two Interfedtions Y, 7;
fall between 4 and B, and confequently ei- ~
ther of the leffer Roots is lefs than 7 03
but the third and greater always exceeds 3 b;
but is excecded by %, But if the Centét |
|
Aa .7 - 7
— Mifcellanea Curtofa. 112
falls in the fpace G NU, there are two
ane ' tg i i
greater than 4 6, but lefS than ; 4, but the
third is 8. the two others, and confequent~ :
ly lefs than 3; but uling the Limitation o4
the Quantity p, the Roots are included in
narrower Bounds. For the greateft Root 1s
eee
‘defS than Vv Pen a pk 3 b, but greater
than Vie b2—p-\-353 but when 3 0d is lefs
rials / i} i hr ee /
than p, that Limit becomes \V § 62-3 p13 4.
oes
‘The mean Root is always lefs than / AUT.
“+4, but greater than }b6—‘V 42-4 ;
but the leaft Root never excéeds this Limit,
but vanifhes with the Quantity ¢.
In the fecond Formulz, according to*the
prefcrib’d Laws, there are two Afhrmative
and one Negative Root; and the Center fal-
ling in the Space G PE, one of the Affirma-
tives is greater, and the other lefs than 4 @,
but the greater exceeds not 6; but the Ne-
gative cannot be greater than V4 bb—— b,
and it is the difference of 6 and the Sum of
the Affirmative Roots. But the Center be-
ing pofited in the Space ENGa, either of
‘the Affirmatives is greater than 3; 6, but lefs
than / 3 bb 36; but the Negative is ever
Vefs than } 4. But the nearer Limits (from
the Quantity p given) aré V/ 2 UU oe Y,
of the preateft Afhrmative Root; than
Which it”.is always lefs, as alfo greater
F J than
114. Mifcellanea Curiofa.
“than ‘VE bb—§p-L-$b5 yet the other Af
firmative Root (which i is diminifh’d with the
Quantity 9) is lefs than this Limit. But.
the “Negative Root is always lefs than -
V3 bb -\- $p— 3b, and the Quantity q be-
ing wanting, vanifhes.
In the third Formula, theperd are two Ne-
gatives and one Affirmative. In this, as in
the fourth, the Roots are not limitted by
the Quantity 6. But the Affirmative Root is
Doaeaecereee
ever lefs than V4 bb- $bb--+p-+- 3, yet grea-
ter than “f pi pa Kb 4} bs; and the greateft
of the Negatives is always greater than
V3 bb-LS p—%b, but lets than V poi bb
ae 2b. But the lefs of the Negatives: is al-
ways. leffen’d with the leffen’d Qantity ¢.
In the fourth Formula, the Center falling
within the Space La P D 5 if there be two
Affirmative and one Negative Root, the
greateft of the Affirmative Roots cannot be
greater than von p+4 x bb “+2 4, nor lefs than
V3 9 be +3 tp; 4. But the lefs Negative
leis than) wea oe
ae $bb-+-4p—}6, and greater
than V p-+-ibb—ib. But ’tis to be noted
here, that ‘the Negative Roots are every
- where mark’d with the Affirmative Sine, be-
caufe thefe are the Affirmative Roots of thofe
four Equations, in which is found ++, andq
= 1s
| Mifcellanea Curiofa. . 118
‘5 affeGted with the contrary Sine; as I in-
timated above. © | :
The Demonttration of all thefe things fol-
lows from hence, that where-ever the Cen-
ter of the Circle R falls upon the Curve
Lines UPX or UAL, the Circumference
of it touches the Parabola in a Point whofe
diffance. from the Axis is / } V H, and cuts
it on the other fide the Axis at the diftance
of 2./}U H; but when the Center falls on
me tiie 17 PD, one of the Roots is =o;
and confequently the Cubick Equation is re-
duced to a -Quadratick one, or to 22— bz
--l- p ==0, the Roots of which give the Limits
when the Quantity 7 vanifhes; and by how
‘much the lefs g becomes, by fo much the
nearer do the Roots approach to thefe Li-
mits. The Equation is alfo Quadratical;
when the Center falls in the Axis; that is,
‘when 2 9 = ¢ bp — 3; 63, in the firft Formula ;
2 9==4, 03'— 2bp,in the fecond; in) the
third ’tis impoflible ; but in the fourth, when
2 9 = 27 03 --% bps in which cafe the lefs of
the Affirmative Roots is ; 6, and the greater
Vv 3 6b -+p-l 3, but the Negative V 3 be
a 40. In. the firft Formula, the Roots
are 3b; and 30+ / 30) -—p. But in the
fecond, the Affirmatives are tb, and “/ ; bb
—p-|- 3 6, but the Negative V/ Obs peer
And thefe things may feem to fuffice in Cu-
bicks ; but becaufe of the excellent uft of the
Method, by which, by the help of the T: ey
a ee ee Se | of
~
“*
6 Mifcellanea Cuviofa.
of Sines, the Roots of thefe Equations are” :
a
i
N
found ; I thought convenient to add an Ex-
ample or two, by which the Compendium of
that PraQife may be rendred manifeft. Let :
the Equation 23 — 3922-|- 4792 — 1881 = 9,
be propos’; and the Roots z are fought.
3 j eREE TG / i ig
Here V.5.60 ~ 3). Vee Nar , whofe
double / 37 “4 is the Radius of the Circle;
alfo 3763-1549 —% bp=2197 9405 — 311325
Vidz 5 9% V 9%
24. ;
or ——--— is the Tabular Sine of the Angle 3_
_ 9 V 93 |
that is, making a Divifion by the help of
the Logarithms, Log. 9.9251560, to which
correfponds an Angle of 57°.19.11,. The
third part of this is 19°. 6. 24’. and of the
Complement, is 40°. 53. 36. The Sines
give the Logs, 9.514983 and 9.816011 35.
which multiplied into the Radius Veer oS
produce Y& and Y&, Log. 0.301030 = 2,
and Log. 0.601059 = 4, but the third Y&
is equal to the Sum of them, or 6& And
therefore the Roots are 13-4 =5393 13-72
— 11, 137619; @ which feveral ones
the foremention’d Equation is compos‘d.
Where ’tis to. be noted that the two leffer.
Roots, do not exceed } 6 or 13, becaufe the
Center R in the Conftruction falls on the
right hand of the Axis that is, ¢ Op is. lefs
than 3y'b3 4" 2 4. : Se fis
3 For
Mifcellanea Curtofa. 117
For another Example, Jet us enquire out
the Roots of the Equation +3 — 1512 — 229~
ssubpees Flere, V$ bb tp = V-1013=
/ d, and the Radius of the Circle = V 405 +.
Alfo 27 63 -T¢ op 1391251 5722 1 2622
V ddd ro1tV tort
the.Tabular Sine of an Arch, whofe Log:
9.9736426, and the Arch it felf 70°. 14°. 22”
The thitd part ‘of ‘it, is 23°. 24’: 47°34, and
of the Complement, is 36°. 35. 12 3, whofe
Log. Sines are 9.999183, and 9.775275, to
which adding the Log. \/ cost, we have the
Log. 0.903089==8, and. Log. 1.079181==!2,
the Sum of which is equal to 20. Hence we
ponelude that 20 1-46 or 25, is equal to
the Affirmative Root, and 8 and 12 —3 4,
that is 8 and 7 equal to the Negative Roots.
But if the Equation had been +3 + 15+2—
22.9x—§25==0, then 8 & 7 had been the Afhr-
mative Roots, and 25 the Negative. As for
the other Cubicks which are explicable by
one only Root, they are to be refolv’d by
Cardaw’s Rules, after the fecond Term is
taken away; neither do I fee how the bufi-
nefs can be done with lefs Calculation.
But if this Root be defir’d to be exprefled.
in the Terms of the Quantities b, p, q, I
fay that in the firft Formula it is, 4.6-\- or —
the Sum or Difference. of the Cubick Roots
D eeetaemeanal
of V5 qq — 405 p2 b2-4, b3 gq bpg tS,
B73 le ote oop (viz, b ifs, 03-2 9 be
a — I 3 ereater
/
118 oo ifcellanea Curiofa,
greater than ~ bp, otherwife —) the Sum,
when 4 6b is greater than p, the difference —
‘
4
‘
when lefs. And in the other Formula, the ©
Root is always compos’d of the fame parts,
only the Sines -\- and — being varied, as
they will eafily perceive that are willing to
make the Tryals. ;
But thefe Roots are readily enough found
by the help of the Log. Table of verfed Sines 5
viz. if the Coefficients are furd or broken
Nambers, and the Roots not to be exprefled «
in Numbers, as moft commonly it happens.
Now this is the Rule. In the firft and fe-
cond Forimuia, if 7 bb be lefs than py tet 3?
—bb=d, and putting the difference be-
tween ~ bp, and 4, 63 4-4 q (that is HR) in
the firft Formulz, and the difference between
%6p-\ 3g and 4, 63 Gin the fecond Formula)
Radivs, let the Angle, whofe angent is
av d, be found. Then, as the €o-five of this
Angle, to the verfed Sine of the fame, fo
the Difference made Radius, to a fourth Quan-
tity, the Cube Root of which will be had by
taking the 3 of its Log. Then dividing 3 p —
5 6b by this Cube Root, let the Divifor be
fubftra&ed from the Quotient, the Remainder
will be the Quantity Y& at Fig.1. The Sum of
this Remainder and 56 will. be the Root,
fought, if the Center- falls on the Right
Hand, of the Axis; otherwife their Difference
will be the Root. But if 3 bb be greater
than P, making HER Radius, let-dV.d, (or
the. diftance of the Paraboloid from the
Axis) be the Sine of fome. Arch: let the
verfed
Mifcelanea Curtofa. 119
verfed Sine of this be multiplied into Radius
Por % ep a, 0339, and taking 3 of the
Log. of the Produd, its Cubick Root will be
obtained, by which let 5 65 — 43 p be divided.
I fay, that the Swm of the Quotient and Di-
vifor, after the fame manner added to or
taken from 3; 6, will give the Root fought.
And the like for the third and fourth £0r-
mula, unlefs that 3, 3+ ¢ op 1-74 is to be
taken for Radius, and § bb \- 4 p into Vs bb
% ip “Or and, for the Sine. But thefe
Rules will be perhaps better underftood by
Examples.
Suppofe the Cubick Equation 23 — 1722 -F
B42. 450 = 0, and let the Root x be
fought. Here } bb is is greater than p, but
q is bigger than the Cube of 30, and there-
fore “tis explicable by one Affirmative Root
alg. Mew ~2- 3* is d>-and *57 “/222 is
to be taken for the Sine, to the Radius a ias
eee 433s that is *3773 and the Arch
agreeing thereto is 15°. 30. 49. The Log.
verfed Sine of this 8.5362376, added to the
Log. of the Radius 2.3095913, makes
- 0.8457889, the 3d part of which 0.2819276,
is the Log. of the Cube Root 1.91394, by
which, as a Divifor, dividing *3* or 4, the
Quotient is 7.37281. The Sum of the Quo-
tient and Divifor encreafed by the addition
of 4 6, is the Root fought, viz. 14.9534, &c.
" Having thus’ difpatch’d -Cubick, Equations,
let us proceed to Biquadratical ones. Thefe
have always either none, or 2, or 4 true
| ] 4. Roots,
120 Mifcellanea Curiofa.
Roots, the determination of which depends ©
partly on the Coejficients, partly on the Sine |
aad Adaguitude of the abfolute Number’ |
siven. A general Conitruéion for all thefe ©
(and that eafy I] conceive enough) I have de- ©
livered at N° 188, ,which I fuppofe the Rea-
dear to be acquainted with, but yet the
Figure relating to that Matter, 1 think
proper to bring hither, (Fig. 2.) In the Con-
ftruétion of the | Equation 24— 623 pe2— qx
Nive eget BD==16, AB==7,. 6b, BK—;
or = the ig pene ore 1 AB=1bb, KE
== 4 .bb Td Po AB =e S47) bb 2p, Be
76 Oe Op, and E.G s= Feb oe Bee oie &
Which done, a Circle on the Center 'G. with
the Radius VG D277, will interfect the Pa-
rabola, either in none, or 2 or, 4 Points,
from whence Perpendiculars let fallon DH,
will give all the Roots .z... But that there
may be 4, ’tis evident that the Center of the
Circle ought-to be found fomewhere within’
a fpace from any Point of which, three Per-
pendiculars may be let fall. upon the Curve
of the Parabola; and alfo that the Radius
is lefs than the “greatelt of thofe Perpendi-
culars, and sreater than. the middle one.
Buc that if the Center. be pofited without
this fpace, fo that there can be but one
Perpendicular let fall upon the Parabola, and
the Radius greater than it, or if it be lefs
than the middle one of the 3 Perpendiculars,
but, greater than the leaft..of them; : then
there can be but two Roots only, But there
—7
sa no Root at all, when the Raclius VG D2—r
¢ i
®
MifceHanea Curiofa, 1291
is lefS than the leaft of the 3, or than the
one as often as there is but one. Now it
remains for us to inquire of what kind this
Space is, by what Limits ’tis diftinguithed,
and under what Conditions the Radius of
the Circle is lefS or greater than the fore-
‘mention’d Perpendiculars. And firft of all,
we muit fhew how a Perpendicular is to be
Meteora upon .the Parabola. “Let '( Fie: '3.).
ABC be a Parabola, AF its Axis, ‘AV o
the Parameter, G the point from whence
the Perpendicular is to be let fall. Let GE
be drawn perpendicular to the Axis, and
VE be bifected in F, and erecting the Per-
pendicular FH on the fame fide of the Axis,
Jet FH=iGE; I fay that a Circle de-
fcrib’d on the Génter H, with the Radius
HA, will interfet the Parabola in three
points, or one, z, the right Lines GZ drawn
to which, will be perpendicular to the Curve
of the Parabola. But now that there may be
3 fuch Interfections, the Center H ought to ©
be fo pofited, as that it may be within the
fpace included by the Paraboloids (in Fig. 1.)
that is, t that FH may be lefs than s/s, FV5,
or FH? lefs than the Cube of } VF; and
foGE ay yey wiil be lefs than ave: VES,
- that is, the fquare of GE will be lefs than
4¢V E?. Therefore thefe Limits coincide
with two Paraboloids of the fame kind with
thofe which were ufed in Cubical Equati-
ae, but whofe Parameter is twice lef$, viz.
g of the Parameter of the Parabola, that is
a of AV. And therefore it is that very
3 Curve
a Say
° ae a Ta TD RRs Be
3 | paifedie Ciriafie
Curve Line, by the Evolution of which the
Parabolais defcrib’d (as Hugenius has demon-
ftrated) and which, the Line DF (Fig, 2.)
which 1s perpendicular to the Parabola in
the point D, is always a Tangent to.-, But
tHe point P (that i is, that in which the right |
Line DF touches the Paraholoid) is the Cen-
ter of a Circle, which (being defcrib’d with
the Radius DP) coincides with the Parabola
in the point D, or has the fame Curvature
with it, as is manifett. cae
' Having therefore defcrib’d fach Parabod
loids UXP, VNa (Fie. 2.) on either fide
the Axis, tis clear, that unlefs the Center »
of the Circle be placed within thefe Limits, -
it cannot interfect the Paraboia in more than
two points. From whence we may deter-
mine, under what conditions, the Coeffici-
ents of the intermediate Terms are reftrain-
ed, in Biquadratick Equations, that fo there
may be four Roots. And at firft fi ght ° tis
plain that p cannot be greater than ¢ bb,
- €viz. in thofe Forms where "tis ‘i p) nor g
than }, 63. But in General, 7, 6? --% pb--
+9, that isEG. the diftance of the Center
trom the/Axts, ought to be lefs than EH =
a VEF, hat is (becavfe VE=#, bb -—
p) than 4 bb + 3p Vi, b2-- or —p, the
Sines “+ and — being left doubtful, that fo
they may be varied according to the nature
of any Equation; as was fhewn above in
Cubicks. ‘Neither would I be offenfively te-
dious to the Learned on the one hand, nor
deprive Learners on the other, of the Ex-
ercife
Maifcellanea Curiofa. 192
ercife and Pleafure, of fending out thefe
things by themfelves. As for the Limita-
tion of the leaft Term 7, it cannot be found
with the fame eafinefs, and that becaufe, ta
let fall a Perpendicular upon the Curve of
a Parabola, is a folid Probleme, and which
cannot be refolv’d without the folution of a
Cubick Equation. Therefore firft of all let
the fecond Term be wanting, or if there,
let it be taken away, fo that the Equation
may have this Form 24. *. pz2. 9x. 7. mo.
And if it be —7,, it is always explicable. by
- two or four Roots; but that there may be
four, the Center of the Circle ought to be
pofited within the foremention’d Paraboloids,
or that it may be—p, and gg may he lefs
than .% p3 or the Cube of ; p. Then let the
- Roots of this Equation y3.%.% py. $9 ==9,
be gotten, the Quantities p and g having the
fame Sines as in the Biquadratick. And
thefe Roots are found expeditioufly enough
by the help of the Table of Sines. But
having found thofe three y (which are ordi-
nately applied, to the Axis of the Parabola
from the points, where the Perpendiculars
to the Curve of it do fall, viz. YZ in Fiz. 3.)
than pyy — 3y4 of the /effer y will denote the
greateft: Quantity of 7, if it be—~+r, than
-which if r be lefs, the Equation will have
four Roots, otherwife but two. But if it
bel r, it ought to be lefs than 3y4 —pyy
of the middle y, for if it be greater, it can
have but two Roots; at leat) ifr be lefs
than 3y4 — pyy of the greareft y. But if
it be greater than this, the Equation
is not: explicable by any true Rooe at all,
ores" : Thefe
124 Mifcellanea Curiofa.
Thefe fame Limits, are (otherwife expreffed
by the Quantity 4 fy VIZ. 3 VY —y4 in the i
cafe, y4—% gy in the fecond, and y4 + 3 qy
in the third. But it may be, that the ae
Yeffer Quanties y may not be far different from
one pihaedl whence it comes to pafs that
both of the Perpendiculars are greater than
' the fight | Line GA, wiz. witen 41 is greater
than’ 4 p?, but ele than? 4 <p? s phe Ente?
falling within the er contain’d between
the Paraboloids of 3 Figu.t. and: 2.9 Tn thie
cafes ifs it be: Aci, "there can be but two
Roots, y¢ b4 gy of ‘the sible: y being grea-
ter than otherwife none. But if 4 gy—y4
of the /eaj? y be greater than 7 mark’d with
the Sine —, but,r be greater than 3 gy —y4
of the mean ¥, then there will be four Roots;
but ¢wo only, if r be found greater than the
former, or Jefs than the latter. But if in
the Equation it be iy Py OF if it be —p and
4 be greater than ,4, p?, the Equation y’. *.
= py. 4 q- is explicable by only one Root y;
that i is, there can be but one Perpendicular
ouly let fall from the Center of the Circle.
Whence it,may be certainly concluded that
there can be but two Roots only in the
given Equation, the Sum of which, if it be
——r, is increas’d with the Quantity rebut
if it be +7, the Quantity y being abe dnd:
that Quantity r ought to be lefs than y4-3
qy, for if it be greater, the Equation pro=
pos’d is abfurd and impoflible.
’Twould be both tedious and needlefS to
‘yun over all Equations of this kind, fince "tis
evident (from what has been already faid)
to thofe that are attentive, which are Ne-
gative
-
Ait
f ae ¥ aa BS q
| Mifcellanea Curiofa. 125 —
gative and which Affirmative, and that the
Limits of thefe Roots are deriv’d from the
found Quantities y. But for an Example
(which any one may imitate in the reft of
of the Cafes) let it be propos’d to difcover
the Limits or Conditions, under which, there
may be four Affirmative Roots in a Biqua-
dratical Equation. Now this will be as of-
ten as the Center of the Circle G is polited
in the fpace UPK (Fg. 2.) and alfo tr or
the Radius of the Circle is lefs than GD.
Whence ’tis plain, that the Equation here
-concern’d is of this Form, <4 —6z3 + pz2 =
gz -+rmos; and that p cannot be greater
than % bb, nor % pb (in this cafe) than ,*¢ 63
“1375 again, ’tis neceflary that 40b—3 p
in ae bb —% p thould be greater than 1% 43
+-39—% 5p; and from thefe Lémzrs, it will
be manifeft that the Center is contain’d
within the fpace UPK.~ But in order to
the determination of the Quantity x, this
Cubick Equation muft be firft folv’d, y3. x.
will be obtain’d the Points upoa which falt
the Perpendiculars from the Center to the
- Curve of the Parabola. Now having found
the three Values of this y; the Quantity r
eught to be lefs than 2#, 64-1 4 bq —i% bop.
> 3y4 — 3 b2y2-+ pyy of the middle y, but
greater than 236 6414 bg — i, bbp 1 3y4 —
$2 y2-+- pyy of the leaft y. But if r exceed
thefe Limits, there can be but two Roots
_ obtain’d. Laftly, if 2$, b4-+ 4 bp — x, bbp
3y4 —=% bby a pyy of the greatelt Vy be grea-
ter than 7, then the Equation propos’d 1s
im-
126 0 =Maifcellanea Curiofa. — |
~impoflible’ It happens alfo that there aré
four Affirmative Roots, when the Center G
is pofited in the little fpace UTS, viz:
drawing RTS perpendicular upon the mid-
dle of the fuppofed Line AD. But this
comes to pafs when p is greater than ,& 00,
and % bb —}p / i bb —<% p greater than §
pb —1i, 63 —449. In which cafe always two, .
_fometimes three of the Roots are greater
than 4b. | Dar ays ;
But ’tis to be noted here that that Limit
‘produced from the /ea/t y, is fometimes Nes
gative, or lefs than nothing , wz. as often
as the greateft of the three Perpendiculars is
greater than GD (Fig. 2.) If this happens;
the Quantity --r may be dimintfh’d to no-
thing from the Limit prefcrib’d, by the
middle y. But the defect of a Limit from the
leaft y, fhews how great —r may be in the
Equation, if there be three Affirmative Roots
and one Negative one ; which if it exceeds,
‘there can be but two, one Affirmatsve and
the other Negative. And all thefe things
are demonitrated from hence, that the fore-
mention d Limits of the Quantity r, ave the dif-
fevences of the Squares of the Line GD, and the
Perpendiculars to the Curve of the Parabola.
But becaufe of the perplexing Cautions ari- .
fing from the diverfity of Sines with thefe E=
quations, ’tis better always to take away the |
fecond Term, and then to inquire out the
number of Roots and the Sines, according to
the Rules already deliver’d ; efpecially if thofe
Quantities y are not much different from oné
another. But of thefe four Affirmative Roots,
: : : EWwo
ore
et gale 1 Se
ea en a Smarr f > aeneiieendian: ofa oe
es : ‘ r ae 4 eo’ ey
% i ; : ”
> mls 5 : | . ;
é 7 ; | | |
7 . ‘ :
| | 3 L
k , al i /
i -
i _ |
; . | | |
j f 7
Z) $ : | |
{ 7 ; ; :
. F . i
‘ ‘ - : j - - : ; |
} | e
, | |
; | | | |
+
. .
‘ 4
j .
= i
4 i ;
>
A=
aos pl ime
Maifcellanea Curiofa. © 127
two are always lefs than 4 b, and two grea-
ter, viz. if DG be lefs than AG, or % pb than
é, 63-4. But three are always lefs than § 4,
as often as the mean Perpendicular (or that
found from the mean y) is greater than AG,
org bby greater than 3y3-—pyy of the fame
mean y. The fourth and greateft Root is
greater than the greateft y 1-4; and “tis
equal to the difference of 4 and the Sum of
the other three Roots, and therefore is .lefs
than 6. ve
_ But ’tis now time to have done with this
Matter. Perhaps thofe that more perfectly
underftand the Nature of the Parabola, may
be able to do all thefe things after a more
compendious manner. But there is fome
caufe to doubt, whether all thefe Quantities-
6. p, 4.7. can be rightly determin’d without
the Refolution of a Cubick Equation, or no.
For whatfoever is done in Plain Equations in
this Matter, exhibits, not the true Limits,
but fome Approximations only.
Some
128) | Mifcellanea Curiofa, —
‘Bove Tilaftviots Specimens + af thé
Doctrine of Fisscmuen 3 or fixe
amples by wibich 4s clearly
afhewn the Ufe and Excellency
of that Method in folving Geo-
metrical “Problems.
By Ab. De Moivre.
OU have here alfo the Method tk
I promis‘, concerning the Quadratures
of Curvilineal Figures, the Dimenfion of
the Solids generated by the Relation of a
Plane (and of the Surfaces) the Redtification
of Curve Lines, and the Calculation of the
Center of Gravity. I know thefe Points
have been already handled by feveral very
learned Men—- But I hope this At-
tempt of mine will neverthelefs net alto-
gether difpleafe, if (efpecially) I have had
the good Luck to find a fhorter and more |
expeditious way to thefe things, than what
is commonly known.
But before I proceed farther, I would
have it obferv’d, that I make fe here, of
what the celebrated Mr. Newton has de-
monftrated, Page 251, 252, 253. Princ. Phil.
- concerning the Momentaneous Increments. or
Decrements, of Quantities that Increafe or De-
crease
| Mifcellanea Curiofa: 129
creafe by @ continual Flux; Efpecially, that
Le
the Momentane of any Power, as A m™ is
gt 23 |
n ei Ue ngs
em, a AA 7 a. Partners “the Fluxion en
en ae :
-a/4L" being given, the flowing Quantity
7 n
A™ may be found; Firft, By ftriking a
out of the Fluxion;, Secondly, By encreafing
the Index of the Fluxion by Unity ; Third-
ly, By dividing the Fluxion by the Index
thus increafed by Unity: In the following
Difcourfe, we fhall exprefs -4b/ciffe of any
Curve by x, its Fluxion by x, the ordinate
by y, and its Fluxion by y. Thefe things
fuppos’d ; that we may come to the Quadra-
ture of Curves, Firft, Take the value of the
ordinate applicate, by the help of the Equa-.
tion exprefling the Nature of the Curve:
Secondly, Let this Value be multiplied by
the Fluxion of the Abfcifle; for the Pro-
‘du& arifing is the Fluxion of the rea.
Thirdly, Having the Fluxion of the Area,
Tet the flowing Quantity be found, and fo
we fhall have the Area fought. Ex. gr. Let
mn
the Equation x ==y be propos’d, which ex-
prefles the Nature of all forts of Paraboloids.
< 4 mn
The Value of » is «” which multiplied by
: K nia
120 Mife ellanea cuetiha.
‘x, gives a? x st the Fliciog of the Aree
“
and confeanentiy the Area fanght is ——
m+n
jt | =)
x , or Cfubftituting y inftead of W
X Ye
m-|- iL
Again, fuppofe a Curve, whofe Equation
is x? + aaxx yy (which is the firft of the
Excellent Mr. Craig’s Examples) putting a
e Met My ‘aa, the Eluxion of the Area will be
eee ee
we ob ‘aa. Which Expreffion involving
a furd Quantity, let us ice “xe aaTa,
then will xv" aamx*, and confequently
xe mzzZ3 and fabititetioe zx, and z for xx
—st
and oy + an, the Fluxion thus freed from
Surds, will be x” 2: ; which reduced to its Ori-
ginal; 2? and putting V xa-t-aa for z, we have
“2 ya\aa V xx aa for the Area fought.
~ But to fhew more effectually how eafily
thefe Quadratures are perform’d, I fhall add
one Example | more. Let the Equation of the
ag?
v
Curve be : = y*, therefore y = ——,
‘ ae 7 ; one
: : me Varka
et ae : Fae eae Rca ‘ ‘ P e i biog
and therefore ———-_ is’ the’. Fluxion of the
“/ x'\"Aa | Area.
: Mifeellance c ur 0
Area. Put: Moet fam a hence KHL a
Pe by
we
F272 et 2AZe
alte! #t2zz. Therefore
poe
and confequently 3; z>-—242, or ; cf a4
Vela will be the Area fought.
But it often happens that we meet a
fome Curves (fuch as the Circle and Hyper~
bola) which are of fuch a Nature, that ’tis -
in vain to attempt the freeing the Fluxions
of them from Swrds. And then reducing
the Value of the ordinate into an inf-
mite Series, and multiplying the feveral
Terms of the Series into the Fluxion of
the Abfciffe (as before) let the Hiuent of
each of thoie Terms be found, and fo there
will arife a new Series, which will exhibit
the Quadrature of the Curve.
This Method is with the fame cafe appli-
ed to the Menfuration of the Solids generated
by the Rotation 0° a Plane; viz. taking for
their Fluxions, the Produ& of the Floxion
of the Abfcifle into the circular Bafis. Let
the Proportion of a Square to the infcrib’d
Circle be —..- The. Equation. exprefling the
I ; | |
Nature of a Circle is yy = dx 44x; there-
fore 4 dxx -~ x? x is the Fluxion of a Portion
“ : Ki 2 of
4
\
132 : _Mifcellanea Curtofa.
of the Sphere, and confequently the Portion
Yee ees an
= *
rie
it felf is 43 dx? 4 4x3, and the circumfcrib’d
‘ “a
Cylinder is 4 dx? — x3. Therefore the Por-
17 ;
tion of the Sphere is to the circum{fcrib’d Cy-
linder, is ast d—3xtodex.
The Reétification of Curve Lines will be ob-
tain’d, if we confider the Fluxion of the
Curve as a Hypothenufe of a Rectangular
Triangle, whofe fides are the Fluxions of
the Ordinate and Abfciffe. But in the Ex-
preffion of this Hypothenufe, care muft be
taken that only one of the Fluxions be re-
- maining, as alfo only one of the indetermi-
nate Quantities, viz. that whofe Fluxion is
retaind. Some Examples will render this
clear. : ons
(Fig. 1.) The right Sine CB being given,
to find the Arch AC. Let AB=x. CB=y.
OA=r. CE the Fluxion of the Abfciffe,
ED the Fluxion of the Ordinate, CD the
Fluxion of the Arch CA. From the Pro-.
perty of the Circle 2rx—x«x = yy, whence
ari me 24x 2yy, and therefore xi yy. But
at ae a
CDizytaxaytyy mayb
| i rm are ex.
o ret | ita
yyy rryy 5 :‘therefore CD = --—-——-
Ue eestaipsonn ted yg Lb Sa
17 em Vy iT yy Vir py
=
mom
ee eS
Maifcellanea Curiofa. 133
I 4 : y i : i
See yy x my. «And
Vir=yy |
confeqnently if rr —yy be thrown into an ine ©
“finite Series, and the feveral Members of it
be multiplied into ry, and then the flowing
Quantity of each be taken, we fhall have the
length of the Arch AC. After the fame
manner, giving the verfed Sine, the fame
Arch may be found. For refuming the E-
quation found above 2rx— 2x = 2yy, we
have y Fire wx, but CDisxx hy swe
Ys
TKK we WX RN NM
— ee
Uae ae
rrxn ms 2rexe Te? eK
—g—-———- ————, that is, (reducing all
2re x8 |
to the fame Denominator, and expunging
ha TVX
contradifory Terms) ————~, whence
es 2K mm HH -
CD= re ~— ; and confequently the
PR) RTT ETD ony
“hia M orm xx .
length of the Arch AC may be eafily found
from what is faid already. :
Aut ' K 3 : The
(134 Mifcellanea Curwfa.
The Fluxion of the Curve Line is fome-
times more .eafily found by comparing the
two fimilar Triangles CED, CBO, for this
Proportion arifes, CB: CO: : CD; that is
et
for the Circle NE bye axe: Tt 1K
Af UK ARK.
The Curve of the Cycloid may be deter-
min’d by the fame Method too. Let (Fig. 2.)
ALK be a Semicycloid, whofe generating
Circle is ADL. Having any point as B in
the Diameter AL, draw BI parallel to the
Bafe LK meeting the Peripheus of the Cir-
cle in the point D, compleat the Rectangle
AEIB, and draw FH parallel to EI and infi-
nitely near to it, as alfo BI cutting FH in
G, and the Curve A in -H. Pot AL ed.
AB(= El) =x. GH= x. It is known that
the right Line BG is everywhere equal to.
the Sum of the Arch AD and the right Sine
BD; whence ’tis manifeft, that the Fluxion
IG is alfo the Aggregate of the Fluxions of
the Arch AD and the right Sine BD. “But —
the Fluxion of the Arch AD, was found
sdx and the Fluxion of the right
————
Leal ne eae
a ep co wae
Sine BD will be found to be. dic m 2a
| Oe ge el
therefore - IG = Mee hm 0 and therefore
aes
Ndi Hy | ee
— Mifcellanea Curwfa. 035
| TE] ( sia Gil) = despots Meals ; from
dm xx
Baad ait oT. foe
de eX x
pl»
4
and confequently Al m2d« =2v/d«x=2AD.
This Conclufion may alfo very eafily be .de-
daed from the known Property of the Tan-
sent... For fince the little part of it, as 1H,
‘5 always parallel,to the Chord. AD, the
Tclangles IGH, ABD are fimilar, ~whence
AB: AD:: GH: IH, that isa Vda ck:
ergy) 3 wort (a1! ane
xJde, therefore IH maV/dx dx x. By
oe ae
the help of the Fluxion IH alfo, we may
inveftigate the Area of the Cycloid. The
Fluxion of the Area AKI, is the Rectangle
HIG dex — xx. xVde a xx. But the
Fluxion of the Portion ABD is the fame;
therefore the Area AEI and the correfponr
dent Portion (of the Circle) ABD, are: ae
ways equal. RELUISe iW BGA
a , a cu
W's
1360 Mifcellanea Curiofa, =
Let AB (Fig. 3.) be the Curve of the Pa-—
rabola, whofe Axis is AF, Parameter a; let
AE= x, EB=y, AB=x, BD=x, DCxy,
BC=x. The Equation exprefling the Na-
ture of the Parabola, being ax =yy, we have
ee Wy whence x ayy 5 but BC? = BD4
i a
eo
2
1 tph, thie i'd eae eee
aAa-
ayy Te aayy, and therefore Pe 7 V ay? + ad
. - :
Aaa 7 Bs
=y Vy --iaa. If now by this Expref-
z
2 4
fon y Vy + 4 4a be thrown into an infinite
Series, the Curve AB will eafily be known.
It appears farther, that giving an Alyperboli-
cal Space, this Curve is alfo given, and wice
-werfa. For } az cy Vy -- 4 aa, and confe-
quently 3 4x is the Space whofe Fluxion is
el y* -\'G aa.” But fach.a Space is no other
than the Exteriour ' (Equilateral) Hyperbola
ABEG, whofe Semiaxis AB =4 4, its Ab-
- fcille AE =y, and its Ordinate EGmx.
For
Mifcellanea Curwfa, 137
For the Menfuration of @ furface defcrib’d
by the Converfion of a Curve round its Axis>
we are to affume for the Fluxion of it, a Cy-
lindrick Superficies, whofe Altitude is the
Fluxion of the Curve, and whofe diftance
from the Axis is the Ordinate Applicate cor-
refponding to that Fluxion. Ex. gr. Let AC
- be the Arch of a Circle, which turning round
the Axis AD, generates a fpherical'Superficies,
which we would meafure. Now DC the
_ Fluxion of the Arch is already found to be
rx
= ————, which if we multiply by the
pune ;
Periphery belonging to the Radius BC, that
is, by £N are xx (putting + the Ratio of
the Circumference to the Radius) we thall
have cx for the Fluxion of the fpherical Su-
perficies, and confequently that Superficies it
io, is tx. . rae
As for Centers of Gravity; having gotten
the. Fluxion of the Solid or Surface, and
_ multiplied the fame into its diftance from
the Vertex, the flowing Quantity muft be
found, which divided by the Solid or Surface.
it felf, the Quotient will fhew the diftance of
the Center of Gravity from the Vertex. Thus
to find the Center of Gravity of all the Pa-
_faboloids; their Fluxion is thus generally
Fa | 3 expreiled
178 a Criofas |
4 im
etpriee ® “ which multiplied by x, , makes
a te
Be i the doings Quantity of which,
* ae ao |
ViX, —a—— KE divided. by Set Area P56
mr-\- 22. i Ch A y
m~\~ 1
Ayla m-\-2n-
the Paraboloid 2) bi BO = x,
- men man
the diftance of the Center of Gravity from
the Vertex.
The Center of Gravity of a Portion of a
Sphere, is found after the fame manaer. }
al
its Fluxion 4 dxx—.x’x multiplied into «,
4
makes 4 dx?x— xx, whofe flowing Quan-.
ap
‘tity 4pae! — 2 ae? Bp he eee
2
Content of the Portion, vz. 4.3.dxx—3 x's
wa :
pa 4% 4d — 3%
gives
3 K—c, OF
2a—3% 6d — 4x
flance of the Center of Pp erie from the
Vertex. 4 : ; ine teh
xy the ‘di-
—— Mifcellanea Curiofa. 139
‘My defign here was not to be large, and
purfue all the Difficulties that may oc-
cur. 7Lis fufficient to have made a be-
ginning, and led the Way to thofe greater
Pranss..s> a
Hee oti 4 MMe-—
149 Mifeellanea Ciriofan
4 "Method dof woe ane fois
of Curves, or Reducng them
to more fi mole Curves.
by A. De Moivre, R.S.S.
ET A be the Area of a Cane whofe
- Abferffe “1s ¥, and Ordinate” Applicate
x ae —xx. Let B be the Area of a Curve,
whofe Abfciffe is the fame with the former,
m=
and its Ordinate x Mdee — xX. Put
Viswas “x =e a4 basse ae “eS aa ad B into
am-|-r.
am-|-4
ah Socae
ae
AON d to 2m-\-1 soe 3 |
- ahi case y =e
z
a at gg 211—I a
— in 10 “ts jin awe a =— 5
—“ 2m-|-1 m—t . ome
got into 2m-\-4, into 5 2m-|-2 218
x ys es eu, dec.
Where we are to Note, 1. That z is fup-
posd to be an ‘Integer and Affirmative
bi oe a. That the Quantity d”B,.in the
| Series
™
Maifcellanea Curiofa. 42
Series exprefled by P, is to be multiplied in-
to as many Terms as there are Unities in z.
3. That as many of the following Series ex-
prefled by -Q.,—-R,—5,—T, &c. are-to be
taken, as there are Unities in. Which_to il-
‘Iuftrate by an Example ortwo: If#=i, the
eth ol See Oo gk
I fay that Ard”B. deben Sela “Tie | 3g
x rt:
and if #=2, then A=a”B into 77) into
2n—2 Ms I tml» a _d. A 2m-t
ee ae oe
“23, 4. That if y be put SV de + xx,
then A will =Q~—-R-+S-—T, &e. AP.
CO ROL. i:
If m be put = to any Term of the follow-
ing Series %, 2,25 2> 2) 2» &c. then the
Quadrature of the Curve whofe Ordinate is
x ae xx, OY X Vf dic -|- «x, will be ex-
prefled in finite Terms, and be found by
our Series. To illuftrate which by an Ex-
ample or two; Let it be required to
find the Area of a Curve whofe Ordinate is
4 V dee — xx. Let us imagine this Curve
to be compar’d with another Curve whofe
< 3 eigen a
= C @ ae e
Ordinate is x os —xx. Now becaufe in
this cafe 2 =i1, therefore will Am d’B into
Qm-i Ll Mem td | i :
geeks mie” 73 But mma ENS
| fore
142 Mafcellanea Curiofa. |
fore 2m-—-1=0, and thesefove A pa
De = =i ays
av x3.
It is here to be obferv’d, that the Area —
thus found, is fometimes deficient from the
true Area, by a given Quantity, or exceeds
it by that fame given Quantity. And in or-
der to find that. Defec& or Excefs, let the
Area found be fuppos’d to be encreas’d or
diminifh'd, by a given Quantity g, and then
putting + = 0, let the ie increas’d or di-
minifh’d, be fuppos’d = Thus i in the pre-
fent cale, we fhall find 7 3.4V/d, and con-
fequently AS 3 afd — 2y>
ans
COROL. I.
If z be put equal to any Term of the
following Series; 3574,, 5) 6, 7 Ce tem eee
Quadrature of of the Curve whofe Ordinate is —
7
x Vide — xx, — xx, Or x ii alist =X, 1S expref=
fed in finite Terms, and is eis. ’by our Se-
ries.
Let the Area of the Curve he to be
found, whofe Gidina td is “x ego oa Ce
Suppofe it to be compared with the Area of
‘a Circle, which call 4. Then will m =o,
2 3, and foo AM P-Qo#R-S. But |
fince, in the Denominator of the third Term
‘ by which 4B is multiplied, there is found
mr ick
Mifceanea Curiofa. 142
2m, a Quantity infinitely fmall, or rather .
nothing ; the Quantity exprefsd by P is In-
finite; and for the fame reafon the Quan-
tity exprefsd by —S is Infinite, and {6 the
_ Quantities A, ~Q, —R, do vanifh. There-
‘ fore P=S, and dividing the Equation by
2m-\-1 2m—t1 ° ane
into ———) we have d”B into d
2m-\-4. 2m-|-2 | : ied
dd’ m—3 20-3 Us at.
ic rt 4 3 : 7 ——— > * 3 e
mow. “y?, or 4B into —~ mddx
and putting o and 3 for and #, there will
be dB into -2=y3?, or B= — 2y?.
“P gx03
COROL. III.
If m be put equal to any Term of the fol-
lowing Series, —2, 1, 0, 1, 2) 3945 5, &c.
the Quadrature of the Curve, whofe Ordi-
nate is « V ade —xx, depends upon the Qua-
drature of the Circle. But the Area of the
im” Papen pam ene
Curve, whofe Ordinate is + V dx-Lxx, de-
pends upon the Quadrature of the Hyper-
bola; and the relation of that Curve to the
Circle of the Hyperbola, is exhibited by our
Series in finite Terms. |
‘COROL.
Mw:e
TTS ER nS aa ate eR
144 Maifcellanea Curiofa,
COROL
mail.
’
“If m be expounded by any other Term; |
different from them already mention’d; the
Curve whofe Ordinate is *, Viasat: or
cc ene : i rig Ar
x NL die + xx, 18 Neither exadtly fquar’d, nor
does it depend upon the Circle or Hyper-
bola, but is. reduced to a more fimple Curve
by our Series. |
: THEOR. If
Let A be the Area ofa Curve whofe Ab:
re
fciffe is x, and Ordinate Vde — ax. Let B
be the Area of a Curve whofe Abfcifle is
the fame with the former, but the Ordinate
m—n ; ;
x :
NI dex mixx Let Vda xe =, | Then will
Mifeellanea Curiofue 145
: a ae. M3 C= 2m—5 .
A=d"B into into into cn) into
| 2m 2m——2 -2m—s.
2m—7 =
2m—6 SS
“2 m—r
—_-— NX ee vet
in 7
| ae 2M—1 Mee
——— Into Ls <= R.
m—
dd a 2° mt
a ee nto y=
RO 4 | Loe ae 2M~——3 . 2m—s5
— .— Into ——— into into
m—3 2m 2m—2 2m—4.
m
Pe of “y=eT, XC.
| The Obfervations to the firft 7 heoren, take
‘place here alfo, as in what follows.
GOROL
if m be put equal to any Term of the
following Series, 3,3, 2) 3.2, &c. the Qua-
drature of the Curve whofe Ordinate is
- mM ‘ m
ive x
———, or ————-, is exprefled in
! MV de — 0 V dex —|- 2"
finite Terms, and exhibited by this Series.
7
boo in oo COLO L,
146 - Mifcellanea Curiofa.
COROL. I.
If x be put equal to any Term of the fol-
lowing Series, 1,2, 3,4, §, 6,7, &c. Every
; x
' Curve whofe Ordinate is ———, OF
af dum
——
ie ® ® e 6
—————, is fquared by this Series in finite
MV docs
Terms.
GORE ee
If m be expounded by any Term of the
following Series, ©, 1, 2, 3) 4) 5) 5) 7, &C.
m
: : x
the Curve whofe Ordinate is —————-, de-
AN mK
pends upon the Quadrature of the Circle.
m
xe
But the Curve whofe Ordinate is ant iy
| Video 4 sie
depends upon the Quadrature of the Hy-
_ perbola.
For if on the Center C, (Fig.20.) the Diame-
ter AB=d, the Circle AEB be defcribed, and
AD be taken =~, alfo erecting the Perpendi-
* cular DE, the Line CE be drawn. Then the
: Se3
| — MifceHanea Curiofa. 147
Sector AEC divided by 34d is equal to the
Area of the Curve whofe Ordinate is
saa “
——emn—m——-, After the fame manner, if on
Wiad tiie’
the Center C, and the Tranfverfe Axis
AB=d, the Equilateral Hyperbola AE be
defcribed, and taking AD-==., and eredting ©
DE at right Angles, and joining CE, the
Sector ACE divided by {4d is equal to the
Area of the -Curve whofe Ordinate is ©
0 j F
(oe ea © Rea =
i ¥ A ~
COROL. IV.
If m be equal to any Term, that does not
fall into the foregoing Limitations, then the
| wn :
Curve whofe’ Ordinate is ————,_is_nei-
) ; : V dic-beexe , .
ther exactly fquar’d, nor does it depend up-
on the Circle or Hyperbola, but is reduced
to a more fimple Curve.
1; Z THEO:
148 - Mifcellanea Curiofa.
THEOR. TL
Let A be the Area of a Curve whofe.
Abfcifle is x, and its Ordinate Applicate —
ve a ‘
x MV rr—xx 5 let B be the Area of a Curve
whofe Abfciffe is alfo x, and its Ordinate
2d | |
x VT —'XX »
occ
aoe!
Let Vir—xx—y. Then will A=/"?B in-
ce 5s 26h
to | i into into’ inte 1 xe a
Mt |
re Oy ae
m\-2 Ha — ems C7. :
Tx: w—t M—32 7
ee a into 2 x —R
74 ti—-2 —3 m—
a into —— into x y3 =—5
: 2 2 a eakts =
&c. peer
ae he 1
G.0 B.C.
If m be expounded by any Term of the
following Series,-1,3, §; 7, 9, &e. the Qua-
drature: of the Curve whofe Ordinate is
ij ————— YL pment
at ae or x Nils la is had in finite
Terms, and that by the help of this Theo-
rem. a
e q
we tea ke he
COROL
| Mifcellanea Curtofa. 149
£0 ROL vit
If m be expounded by ae Term of the
following meries, 2-49, 4, 5,6, &c. then the
oy
—~
_ Curve whofe Ordinate ts x Nye be or
io
ele aex: Coes is exactly {quar'd by this
' ‘Toren.
COROL. IF.
If m be expounded by any Term of the
following Series, — 2, 0, 2,4, 6, 8,&c. then
_ the Quadrature of the Curve whafe Ordinate
piscx lk ee, depends upon the Circle, but
the Quadrature of the Curve whole Ordinate
iS x TA si dea |-aeee, ee upon the Hyper-
bola.
COROL.: IV.
If m be expounded by any Term differ-
ent from thofe already taken notice of;
at
then the Curve whofe Ordinate is + Weg
| m nn:
or x Vert wx, depends neither upon the
_ Circle nor the "Hyperbola, but is reduced to
amore fimple Curve.
O35) 3 THEO-
150 © ~ Mazfcellanea Curiofa.
THEOR. IV.)
Let A be the Area of a Curve whofe Ab-
: i . 7/3 :
: : iis
fciffe is ~, and whofe Ordinate is Rae
Se
let B Lote the Area of a Curve whofe Abfcifie
M—27
x
is alfo x, and its Ordinate —-——. Then —
| Vee
° e N—=f e m— e ; / Naw
will A=r2"B into —— into "= into et
m m—A m—Z4
6
rt m—!I
poe a y= 2.
¥ M—1 3
— into —— « yu—R.
r4 Nn .
———— into — —— : into’ x aa!
m4. m m—2 ae ils
76
® M—— tT ry } wi--- ° Y 4 aaa estates tbat
me ——— Into —— into.—— into ssi a ’y
bs ie Mio Mis ag els ied m—4. |
=— T. ke.
COROL.
— Mifcellanea Curtofa. 151
PeCOROL 1.
If m be expounded by any Term of the
following Series, 1,3, 5, 7, 9, &c- the Qua-
drature of the Curve, whofe Ordinate is _
: m
a,
——, is obtained in finite Terms by this
i a
Theorem.
£0201. i.
_ If » be expounded by. any Term of the
following Series,.1, 2, 3) 4) §, 6, &c. the
—= 27
Fe,
or
Curve, whofe Ordinate is
Cae
is {quar’d exadly by this Theo-
—2n
Pe aa :
P)
| Vereen |
COROL. IIL
If m be expounded by any Term of the
following Series, 0, 2, 4, 5) 8 10, &c. the
“Quadrature of the Curve, whofe Ordinate
meds : L 4 : 1s
is2 = Mifcellanea Curiofa.
\ ae uff . ; :
is ——--_ depends upon the Quadrature
eS | | ed
of the Circle. For if on the Center C, and
the Radius CA=r, the Circle AEG be de-
ferib’d, and taking CD==*,. DE be erected
perpendicular to CD, :and.CE be drawn ;
then the Sector CAE, divided by v5 7%. (187
equal to the Area of the Curve, whofe Or- |
O-
n | ‘a
dinate is ip tee manner, (Fig.21-) |
id ae |
if on the Center C, and the Scnieventvertey
Axis CA=r, the Equilateral Hyperbola KAM
be defcrib’d, then drawing CF -perpendicu-
lar to CA, equal to x, FE parallel to the ~
Axis till it meets the Hyperbola in BE, and
join CE; then the Hyperbolical Sector ACE
divided by : eis equa! to he Area of the
x
he HR
COROL. “I.
Curve, whofe Ordinate is
If m be expounded ie any Term diffe-
rent from the OTEE ONDER then te Curve, |
x: d
whofe Ordinate is -_—__., is inpaehier ex-
Virx-ex ie
adly
| ~Maifcellanea Curiofa. 153
aGly fquar’d, nor does it depend upon the -
Circle or the Hpperbola, but is reduced to
a more fimple Curve.
THEOR.. V.
Let A be the Area of a Curve, whofe Ab-
: {ciffe is x, and its Ordinate —— ; let B be
ee .
the Area of a Curve, whofe Abfciffe alfo is
ye s
_x, and its Ordinate —-—. Then will A=
f Lomas NO aa
eee ee dx" dn ,
d*B— fey ee ee et ee ccc,
m M—1 m—2
/ ; ax
Let the Ordinate be ——., and then the
| Fel d-\-x
eae ddan 3
Area A will = —~——--— Hj eee eC.
; ™ i— 1 M—— 2
a”B.
2 OR OL:
If m be expounded by any Term of the
following Series, 1, 2, 3, 4, §, 6, &c. the
Quadrature of the Curve, whofe Ordinate |
Bia cM i,
is —— or ——, depends upon the Quadra-
dum X mI
LO ew. ture
»
wassa eer
454 Mifcellanen Cariatth
ture of the Hyperbola. For (See Fiz. 22.) | .
drawing DE, EF at right Angles, take EG
=—<d, and draw GH at right Angles to EF,
and equal to it. Wa£thin the Afymptotes
DE, EF, let an Hyperbola be defcrib’d,
pafling through the point H; which done,
take GK—=~, towards EF in the firft Cafe, and
towards F in the fecond; and draw the Or-
dinate KL. Then the Area HGKL divided
by dd, is equal to the Area of the Curve, ©
x? x? a
or ——. Hence the.
, dex d-|-e¢ 4 oay
Solid generated by a Portion of the Ciffod,
while it turns about the Diameter of the
Generating Circle, is exhibited in finite ©
Terms, fuppofing the Quadrature of the Hy-
perbola.
whofe Ordinate is
THEOR. VL
Let A be the Area of a Curve, whofe Ab- —
el
icifle is x, and Ordinate — : tee Be be:
|x
the Area of a Curve, whofe Abfcifle is alfo
gel —an
x, and Ordinate Then will the
rr—\-x% Seobh
an rex 3 ty) |
Area Boe me ee ee fee KC,
Mormon Mm 3 M—5 |
he r2"B.
COROL.
s +
Mifcellanea Curiofa. 155
COR OL:
If m be expounded by any Term of the
following Series, 0, 2, 4,6, 8, &c. the Qua-
drature of the Curve, whofe Ordinate is
oP og : .
——-, depends upon the Reétification of
yr-|-4N
the Arch of a Circle. For defcribing the
Circle AEG on the Center C, with the Ra-
dius CA= 7, draw the Tangent AK—~, and
join CK meeting the Periphery in E; then
the Arch AE divided by 7r is equal to the
x
Area of the Curve, whofe Ordinate is -
is : rr—|-oce
General Corollaries to thefe fix Theorems.
Very Mechanick Curve (whofe Quadra-
ture depends upon any one of that In-
finite Number of Curves, whofe Ordinates
Gremearcest: teem?
have the following Forms, « M dic bax ,
x x
REET eg Cee : Yl see - UL Pa) ;
ies es 9
i VITNK sy, 8 ae dx
Vide xx |
xt
-) may ‘be fquar’d by thefe Series. It
rr-|-xeF :
may fuffice to illuftrate this by an Exam-
ple. : | 7
_ Sup-*
156 Mife eVanea Curiofa’
Suppofing the Cube of the Atch of a Cir-
cle (correfponding to the verfed Sine) to be
the Ordinate of a Curve, whofe Abfciffe is |
the fame verfed Sine; let it be peguir to
find the Area of this Curve.
Let the Abfcifle be «x, the Gikentar Arch
ers then the Fluxion of the Area is cad fet
the Area be v?x—J. ‘Therefore v3 x2-|-30?
Ug e, whence g == 30? UX. Baik hc
re oo ie SO Auar ae
nae , therefore 7 = - —-. But
a ees ges Byer re
: Re ue dx
(by Theor. 25) : a i —
Wie pues Vives
yar); ; and confequently 4 = ddv?u—3
dv* Vs therefore 74V3—5: doy. There-
fore now we are come to es that the.
Fluent, of the Expreffion 34v” Yo is to be
found. 3 |
Let this Fluent be jdv*y—-+.
Therefore idv’y-[-3dvyy—r—hdo?y. |
And confequently r= aduvy=tddex. &
Let y= ddvxr—s.
Therefore stdpe-t dae caauee
ae And
Maifcellanea Curiofa. 157
303 xx
Romer © Geel
ee 1)
mS Seales} (by Theor. 2.) Therefore now
s=jd?u—jd3y. And confequently the Area
fought, es x td! | jdv7y—jddvx-|-,d3
’ v—id>y.
And pcanena s—tddev= .
Since the Solids and Surfaces generated by
the Rotations of Curve Lines, as alfo the
Lengths of Curves, and the Centers of Gra-
vity of all thefe, do depend. ‘upon the Qua-
dratures of Curves; ‘tis plain, that thefe
are eafily obtain’d too, if they depend upon
the foremention’d Curves.
After I had compos’d thefe Theorems,
and fhewn them to the Celebrated Mr. New-
ton, (as the fupream Judge in all Matters of
this Nature) he was pleas’d to give me a
_. fight of fome Papers of his, by which I find
that he has a long time been Matter of a
Method, by which any Trinomial &quation
(exprefling the Nature of a Curve) being
given, that Curve is either fquar’d, or re-
duced to a more fimple one.
And ’twere to be wifh’d, that he thought
fit to communicate to the Publick, not only
thofe Things which he has relating to thefe —
Matters, but others alfo of his Noble In-
ventions, which are not a fmall Number
neither. And I believe this is not my With
alone, but that of the whole Learned World
befides.
I make no queftion but thofe Learned Per-
op (whofe een in the Ate Prudent
ang
158 Mifvellanea Curiofa. -
and otherwhere, have tended fo much to the
Advancement of Mathematicks) have Me- .
thods nat unlike to this of mine; and there-
fore I afcribe no more to my felf in this
Matter, than only that I found out thefe
Theorems, not knowing whether any Body
elfe had done fo before or no; and reduced
them into fo eafiea Form, that the whole
Calculus relating to them, might be taken
'. in, as it were, at one View. o
But before 1° make an end of Writing,
I think it improper, if (having not had an
Opportunity fooner) I make fome little re-
ply to the Famous Mr. Leibaitz’s Animad-
verfions upon my Series for finding the Root
of an Infinite Equation.
That Excellent Perfon thinks fthis Series
not to be General enough, as not reaching
the Cafes where z and y are multiplied in-
to one another; upon which account he
fubftitutes another Series in the room of it,
which he aflerts is infinitely more General.
But that which led him into this fmall Mi-
flake, I guefs to be this, that he took the
Quantities a, b, c, d, &c. for given Quanti-
ties, whereas they were to be us’d indiffe-
rently, either for given or izdererminate ones.
But I fhall add one Example to fhew that
my Series extends to all Cafes. Let the
Equation be zyz—<? =’. Beas ie
In our Theorem let a=zy, b==0, c= —1,
b=z0, s==1, or rather: let g=yy, hbo, eee. .
| | Co) are,
Then in either Cafe will <=—-+ —-++ —
ray. a pam. iti ig? ‘
-|- teas wea eater &e. °
Ak ea Ny gn Tivo
Maifcellanea Curio fa, BB
Two Problems; viz. concerning
the Solid of Leaft Refftance,
and the Curve of Swifte/t De-
foent.
alv'd by J. Craig.
LEMMA.
Vy 'O find the Proportion between the Re-
_ § fiftance made to the Rectangular Tri-
angle AIG, and that made to the circum-
fcrib’d Rectangle AlGg, while each moves in
- a Fluid, in the direction of the Line IA,
from I towards X. |
From any point B let there be drawn BC
perpendicular to AG, Bb parallel to Al, and
BM perpendicular to AI. Then in bB take
bH = CM? and bE= BC, and thro’ the
| ota .
‘points H and E, draw the Lines HA, EA,
which being produced cut Gg in K and Ff.
1 fay the Refiftance of the Triangle AIG
_ is to the Refiftance of the Rectangle AlGg,
as the Area of the Triangle AKG, to.
the Area of the Triangle AFg. And
alfo, that the Refiftance upon any part
of the Line AG, 1s to the Refiftance
upon the correfpondent patt of the Line
| i. 7 Ag
(160 = =©MaifceHanea Curiofa,
Ag (ex. gr. upon AB and Ab) as the
Area AHB to the Area AEB. The Demon-
{tration of which depends upon a General |
Theorem, which I deduced ey eafily from
Prop. 35. haha P- 324.
COROL, he
_ Let BG, bg, be infinitely fmall parts of ©
the Lines AG, Ag, and let bB be produced
to; Ll fay, that the Refiftance upon BG
(which call e) is to the Refiftance mpon bg
(which call E) as GL”: GB’.
For e: E:: KHgb: FEgb; that is,e: E:
bg x bH: bg x bE (by the foregoing Lemma )
therefore es E:: bH: bE s that is; €:'B:
CM? : BC :: CM? : BC?. iat ea BC?::
GL? : GB? (becaufe of the fimilar Trian-
gles BMC, GLB.) ‘Therefore e: EB: GL":
GB’. Q: E:D.
'GOROL. a8 F
The Refiftance upon the infinitely fall _
part GB, is= GL’. For if all the infinite
GB?
ly finall parts in the Line Ag (as bg) be fup-
pos’d equal, then the Refiftance upon bg,
may be exprefs’d by bg, that is E=bg, ane.
fo: E=GL: Therefore: (by Con. 1.8L:
GL’: GB’, whence e= GL?. Qe Be.
GB?
COROL.
Mifcellanea C uviofa: 1.01.
So ROL? JH.
Let r be the Radius, and c¢ the Circum-
ference of any Circle. I. fay, that the Re-
fiftance upon the Conick Surface generated
by the Rotation of the Lineola GB about
Al, is equal to the Product of cx BM into
r
GL’. For the Refiftance upon that Conick
GB’
Surface, is equal to all the Refiftances upon
the Lineola*GB, that is, to all thee; that
is, equal to the Circumference of the Cir-
cle (whofe Radius is BM) multiplied into-e ;
that is, the Reffftance upon that Conick
cx BM
x e, and confe-
: : cx BM GL?
quently (by Coro/. 2.) equal to — x )
ae r GB.
Surface, is equal to
| : r
PROB. IL
To find a Curve Line, by the Rotation
of which a Round Solid {hall be gene-
vated, that, while ’tis moved in a Fluid
Medium, in the Direction of its Axis,
fball meet with the leaft Refiftance.
(Fig. 24-) a
Let OG, GB, be two infinitely {mall Par-
ticles in the Curve fought, which rould a-
bout its Axis, will produce the Solid of
: | M leaft
162 Mifcellanea Curwfa.
leaft Refiftance. Draw BM, GP, perpendi-
cular to AQ, alfo BL, GN, ‘parallel to oe .
and ON, parallel to BM. —
cx BM *.GL?
. Now ais 5, Reales up.
r x GB’ |
on the Surface generated by the noe of
Cy * GPx ON*
the Lincola GBabout AQ, and :
rx GO? 3
is the Refiftancé upon the Sur yee generated
OG, in like manner (by Cor®3.) And
the Sum of both thefe Refiftances malts be
a Minimum, vit. (we
Cx Divi er cx GPx ON?
-= a Mini-
oe
rx GB? : rx OG?
YAU « :
And confequently i in the Line RS (fo pa-
rallel to AQ that ON—GL) the point G
is to be fought, fuch, that this may happens,
which, fuppofing the points O and B to be
fix’d, will be eafily found by the common
Method, de Maximis Minmis. And pro-
fecuting the Calculus, we fhall come at laft
BM x BL . GPx NG
;
- to this. Equation, —
, eBGt teas eee
aM | BMx BL. | ms
whence °tis plain that ——_ =a con- |
BG*
ftant ‘Ouantity: So that if the Abfeiffe
AM = x, and the Ordinate BM=y, then’
will BL = dx, LG = dy (which I have fap-—
ead "d
Mifeelanea Cunibfe. a. 163 |
: pos’d conftant every where in this. Calculus)
and confequently BG* = dx’ -[- dy’, whence
heist |
deed ie dydy*
be any conftant Quantity, and confequent-
ly (to obferve the Law of Homogeneals) we
ydx a
have =,
dxdx-\-dydy” — dy?
by the Iluftrious 1 iii and the cele-
| brated Fo. Bernoulli.
aoa conftant Quantity. nee a
ee os. I.
te fina the poe of Swiftef Defcent. »
(Fig. 25.)
Let ‘BC, Ci, be 1 Sie infinitely fmall Par-
ticles in the Curve fought. Now this Curve
ought to be of fuch a Nature, that, fuppo-
fing a Body to have' fallen from the Hori-
zontal Line AQ, it may pafs from B to D
in the fhorteft Time. Therefore we are to
find out the Point C (in the Line RS drawn
in fuch a manner parallel to AQ., that the
differences of the Ordinates GC, DE, may
be equal) fuch that this ey come to
Batt.
; Now the Veleeity in C is /LC, and that
BC
in D is /QD; therefore ——— is the Time
M 2 ‘of
(164 Mifcelanea Cuviofa. —
- CD... |
of Defcent thro? BC, —— is the Time of
VQD
Defcent thro? CD (by Prop. 54. pag. 158.
Newt.) Therefore the point C ought to be
BC | CD -
VLE QD
Suppofing ‘the points B and D to be fix’d,
let the conftant Quantities GC=DE=m,
LC=b, QD =p; the Indeterminate Quantities
BG =u, CE =z; whence '
| Vn? tw? Vn Le ca Mi
ith that
may be a Atinimum.
a/b Se
@imum. Therefore.
udu rae
ee — 0.
vbVme tut pV? 2?
But du =—dz (becaufe y+ is conftant)
therefore }
u TNE, Le
\ amt
—
5 whence ’tis
Vb AY ig? but lp N/m? ie | i
manifeft that — = a conftant
VON me be
Quantity. Now let the Abfcifle AL= 14
the Ordinate LC=y, and to BG =dx,
6: Bin ge “oe GG
Ba
ot
— Maifcelanea Curiofa. 165
GC =dy, BO=Vdx?+dy?, and let a
be any conftant Quantity. Then hall
aa Cee
dy Vide? + dy?
V dx? + dy?. - But now in all Curves, ’tis
== —, whence dx Varivy x
dw: Vax? +dy? :: as the Subtangent, to
‘the Tangent. Therefore the Nature of the
-Gurve fought is fach, that its Subtangent, 1s
to the Tangent, as /:4/y, which that it is
a Property of the Cycloid, is known to all,
that know that the Tangent of the Cycloid,
is parallel to the Chord of the Conterminal
Arch, in the Generating Circle, whofe Dia-
meter is 4, and whofe Vertex is downwards.
‘And with the like eafe, I can find the
Curve of the Smifteff Defcent, in any other
Hypothefis of Gravity.
M 32 T he
166 Mifcellanea Curiofa.
T be Quadratere of ihe Logarith-
mical Curve.
(Fig. 26.)
By J. Caty. |
ET ONF be the Logarithmical Curve, |
whofe Afymptote is AR, in which let ;
fach.a point A be taken, as that the firft
Ordinate AO may be equal to the Subtan-—
gent or Unity. ’Tis requir’d to find the
Area of the Curvilineal Space AONM com-
prehended under the two, Ordinates AO,
MN, the Abfcifle AM, and the Curve ON.
From. O draw OB, parallel to AM and cut-
ting MN in £3 1 fay, that the Rectangle
under the Segments ME, EN, is. equal to
the Space pene, Demonhratiin.. Let” tie
Ordinate MN =Z, Subtangent AO or ME
m=s; and to the Axis AR let another Cutve
HGE be conftructed, whofe Equation fhall
_ he 2szx7, its Ordinate GM being =~. I
fay, that this Curve is the Quadratix of the
Lonanchenied Curve-(according to the Prin-
ciples of my Method) wz. its Subnormal is
refpectively equal to the Ordinate of this,
as is plain from the Calculus of that Me-
thod. " Therefore (according to what I have
thewn in another place) if to the point G
we draw GC perpendicular and equal toGM,
as alfo HD parallel to GC, and meeting
the Lines GM, cM, in B and D; then he
the
— Mifcellanea Curiofa. 167
the TrapeziumGBDC = AONM. ButGBDC
= GMG-— BMD = $x? ~;BM?= SZ—
2 HAI; but HA = 2 AO? from the Nature
of the Curve HGQ, therefore GBDC = SZ
~ AO1= AOx MN — AOT= AOx MN— AO
SMEx*x MN=ME=SMExEN. _ There-
fore allo AODNM=MEx EN. Q:E-:D.
When I applied my Method to thefe fort
of Figures, I found that a Miftake had
fome way or other crept into M. Bernoulh’s
Calculus. For in his moft excellent Trac&
of the Principles of the Differential Calculus,
he affigns to the Figure whofe Equation is
eS bse Ue. VDE yy
ay, this for its Quadrature, vz. eer?
a ae
ie aad: 8 aly yy
whereas the Area of that Figure 1s Re
Ve eS BAR ae ae, ee 4a
where y denotes the Abjfcifle and z the Or-
dinate. we ; . , AY
ei
Ee IV OE
cee oe ee Se
. ‘ of Ps
To Cah h J rey! § “8
‘esi < ee ee
teat Sa Bae : | :
eat: ae ¥# es .
| imi M ue AA 1 heos
tert Don i ct) Sti
Sagit
vet ySe *
168 . Mifclanea Oxted. @
A Theorem concerning the Propor- —
tion of the Time that a beavy
Body [pends in defcending thro’
a right Laine joining two given
Points, to the (fhorteft) Lame,
in which it paffes from the one
to the other of thefe Points, by
the Arch of «a Cyclord. a
THEO RE ae.
(Fe. 29)
N the Cycloid AVD, whofe Bafis AD is
parallel to the Horizon, and the Vertex
V turn’d downwards, if from A be drawn
the right Line AB meeting the Cycloid in
wny point as B, from whence is drawn BC
perpendicular to the Curve of the Cycloid
in B, and AC. be let fall perpendicular to
BC from the point A: Then the Time that
a Body at reft in A, fpends in defcending
thro’ AB (by the force of its Gravity) 1s to
the Time that it fpends in falling thro’ the
Curve AVB, as AB to AC. {he ha
Thro’ B draw BL parallel to the Axis of
the Cycloid VE, and BK parallel to the Ba-
_ fis AD, meeting the Axis in G, and the
Circle (whofe Diameter is EV) in F and H,
and the Cycloid gt felf in K: Draw the
right:
Mifcellanea Curiofa. 169
right Line EF, which from the Nature of
the Cycloid is parallel to BC; whence BM.
is = EF, and EM= BF= the Arch VE
from the Nature of the Cycloid , and ‘con-
fequently AM is = the Arch EHVF.
By Propofition 25. Part Il. . Horolog. Ofcillat.
Hugen. the Time in which a Body at reft
in A defcribes the Cycloidal Arch AV, is
to the Time of Defcent thro? EV, as the
half Circumference to the Diameter. :
And (by the laft Propofition of the fore-
mention’d Part) the Time of Defcent thro’
VB, after the Defcent thro’? AV (which is
equal to the Time of Defcent thro’ KV,
i the Defcent thro’ AK) is to the Time
of Defcent thro’ AV, as the Arch VF, to
the Semicircumference ; and confequently to
the Time of Defcent thro’ EV, as the Arch
FV, tothe Diameter. Wherefore the Time
of defcribing the Curve AVB, is to the
Time of Defcent thro’? EV, as the Arch
EHVF, tothe Diameter EV. But the Time
of Defcent thro’ EV, is. to the Time of
Defcent thro’? LB or EG, as EV to EF.
Therefore (by Equality) the Time of de-
fcribing AVB, is to the Time of Defcent
‘thro’ LB, as the Arch EHVF, to the Sub-
tenfe EF, that is, as AM to MB. Again,
the Time of Defcent thro’ LB, is to the
Time of Defcent thro’ AB, as LB to AB.
Therefore the Time of defcribing AVB,
is to the Time of Defcent thro’ AB, in the |
Ratio compounded of AM to BM, and LB
to BA, and confequently is equal to the
Ratioof AMx LB to MBx BA. ~ :
But
170 = Mifeellanea Curiofa.
But AM x LB=MBx AC; and there-
fore the Time in which a Body at reft in
A, thal defcribe the Cycloidal Arch AVB,
is to the Time of defcribing the right
oe AB, as MB x AC to MBx BA; that
> as AC to AB. Q: EB: Dis
fans the Demonftration will pines in
dike manner, if the point B be between A
and a i
: Mifcelanea Curiofa. | 17%
An Extrait of a se eh from the
Reverend Dr. John Wallis, to
_ Richard Waller, E/q; Secretary
to the Royal Society, concerning
the Spaces in the Cycloid, which
are perfectly Quadrable. |
| Oxford, Aecae 22. 1695.
sh8 Tuy.
Find it is thought by moft, that there is
no other part of the Semicycboid Figure _
- (adjacent to the Curve ) that is capable of
being pepmetrically Squared, but thefe two,
AR
. The Segment AbV, (Fig. 28.) taking
A pee = Aa, (which was. firft obferv’d. by
Sir Chriftopher Wren, and after him by Huge-
nivs and others) and it is = 43 Key R*
3°
¥ as 1 Des Trilinear 4 dD (taking dD, in
the Parallel d DC, pafling through the Cen-
ter C,) which is = R?.
. But, it is otherwife (as I have fhewed in
my Treatife, De Cycloide, and that, De Motu ;
x the Figures of which latter I retain here,
fas far as they. concern this Occafion ) there
. | being
172 4 Maifcellanea Curiofa. —
being other Portions of it, equally capable
of Quadrature.
In order to which, I there fhew (De AZotu,
Cap. 5. Prop. 20. A. p.802,803,804-) that not
only the Cycloid is Triple to the Circle Ge-
nerant, (which was known before) but that
the re/pective Parts of that are Triple to thofe
of this: Which is the Foundation on which
I build my whole Procefs concerning the Cy-
claid in both Treatifes, (and which is not pre-
tended, that I know of, to have been obferv’d
or known by any Body before me:) That is,
bpaA (Fig.28.) Triple to the Seftor Ba A
(taking 6g parallel to Ba) where-ever, in
the Curve 47+, we take the point 6.
1 then fhew, that the Cycloid ts a Figure
compounded of thefe two; the Semicircle
A Da, and the Trilinear A Darb A, lying
between the two Curves 4D. and Adz,
(and therefore, to Square any part of thefe,
is the fame as to Square the refpective part
of the Cycloid.
I fhew farther (Jbidem, pag. 804.) that this
Trilinear is but a diftorted Figure (by rea-
fon of the Semicircle thruft in between it
and its Axis) which being reftored to its due
Pofition (by taking out the Semicircle into
a different Figure, (as Fig. 29.) and thrufting
the Lines 6B home to the Axis, fo as that
BV be the fame point) is the fame with
Ara, (Fig. 30.) (the Parallelograms bf 4B
being fet upright, which in the Cycloid ftand ©
floping; and the Circular Arches 6 2, (Fig. 28.)
becoming ftreight Lines (in Fig. 30.) and the
‘Lines 6 B being, in both, equal to the refpe-
| ctive |
: Mifcellanea Curiofa. 173
étive Arches B A, every where ;) which there-
fore I call Trilineum Reftitutum (the Trilinear .
reftored to its due Pofition, which Figure I
do not find that any before me has confi- |
der’d: ) So that to Square any part of this,
is the. fame as to Square the refpedctive part
of the Cycloid, (or of the Trilinear in the
Cycloid: ) That which in the Cycloid lies be-
tween two Aiches of the Circle Generant
in different Pofitions, anfwering to that
which, in the reftored Figure, lies between
the refpective ftreight Lines.
And therefore 4d D A, =7d 37, (Fig.28.)
= Aa DA pr, (Fic. 30.) = R*. And
AbkdA, rbk dr, (Fig. 28.) = Abkd A,
Tbk S7, (Fig.30.) —sR. And bk.d (Fig.28.)
= bkd, (Fig. 30.) == R? — sR, Ibid. Cap.17.
B. pag. 756. Where, if 6 be taken above
dak DC, (pafling through the Center C) thefe
Figures are within the Cycloid, and within
the reftored Figure; but without them, if
___b be taken below that Line, and adjacent to
the Curve 467, in both Cafes.
By R, I underftand the Radius of the Cir--
cle Generant; and by s, the Right Sine of
the Arch B 4, whofe verfed Sine is VA.
And, where-ever in my whole Difcourfe
of the Cycloid, or the reftored Trilinear
(which is a Figure of Arches, and a Figure
of verfed Sines) the Arch 2 is no Ingredient
in the-defignation; fuch part or portion of
them is capable of being Geometrically
fquared. But when I exclude a, I do there-
in exclude P (for that is an Arch alfo) and
f=a-+s,ande=—2—s, becaufe a is theres
in included.
: Mr.
174, Mzfcellanea Curiofa. !
Mr. Cafmell, (not being aware that I had
fquared thefe Figures) had done the fame by |
a Method of his own, (which he fhewed me
, lately) which I would have inferted here,
but that he thought it not’ necéflary; and
inftead thereof, hath given me the Quadra-
ture of a Portion of the Epscycloid (which
you will receive with this) and, I think, it
is purely new.
Mifcellanca Curiofa.. 175
The Quadrature of a Portion of
the Epicycloid. Se, GUO
aa — By Mr. Cafwell.
(Fig. 31.)
Supe DPF to be half of an exterior
De Epicycloid, 7 B its Axis, Y the Vertex,
VL B half of the Generant Circle, E£ its
Center; DS the Bafe, C its Center: Bi-
fe& the Arc of the Semicircle Y B in Z, and
on the Center C through Z draw a Circle
cutting the Epicycloid in P: Then I fay the
Curvilinear Triangle YL P will be= BEq
in ae that is, the Square of the Semidia-
meter of the Generant Circle will be to the
Curvilinear Triangle Y LT, as CB the Se-
midiameter of the Bafe, to CE; which CE.
in the exterior Epicycloid is the Sum of the
Semidiameters of the Bafe and Generant,
but in the interior Epicycloid Dp, ’tis the
difference of the faid Semidiameters.
COROLLARY.
In the interior Epicycloid, if CE is } CB,
the Epicycloid then degenerating intoa right .
Line, the Quadrature of the Triangle / xp
will be in effect the fame with the Quadrature
of Hippocrates Chins.
| | CORUL. :
Se RE: Oe hn a eee
TAY ye Fas soars
, id r ‘
bbe,
176: Mifcellanea Curiofa. ane
COROL. IL
Bri
i
If the Semidiameter of the Bafe is fuppofed
infinite, the Epicycloid then being the com-
mon Cycloid, the Area of the faid Triangle
will be equal to the Square of the Radius of
the Generant, and fo it falls in with that —
Theorem which Lalovera found, and calls
Mirabile. | “se :
Though I do not think the abovefaid Qua-
drature can eafily be deduced from what has
been yet publifhed of the Epicycloid, I have
not added the Demonftration; but think it
enough to name a general Propofition from
whence I deduced it, viz. The Segments of
the Generant Circle are to the Correfpondent
Segments of the Epicycloid, as CB to 2CE
-+-CB. For Example, fuppofe Fm G the
Pofition of part of the Generant when the
point F of the exterior Cycloid was defign’d,
then the Segment Fm G x is to the Segment
DFuG::asCBto2CE-|-CB.
And confequently the whole Epicycloid to
the whole Generant in the fame Proportion: —
Which is the only Cafe demonftrated by
Moufieur De /a Hire. :
It follows alfo that inthe Vulgar Cycloid,
its Segments are triple of the Correfpon-
dent Sectors of the Generant, which. was
firgt Shewn-by Dr. Wallise. cee es
A General
ee I Fe, ee ar Pome, AS se t
Ape eae PSIG ae Ue 3 ¥
2 ‘s Mifeellanea Curriofa. oe 77
*
| A General Propofition bering the
— Dunenfion of the Areas in all
thofe kinds of Curves which are
deferib'd by the Equatle Re-
_volution of a Circle upon any |
| Bafis, evther a Rectilineal or a
— Circular one. |
BTS 24 By Edm. Halley.
SFP VIS known that the Primary Cycloid,
= # as alfo the Prolate, and. the Contratted
-one (which they call Trochoids) have been
. largely: handled by the Celebrated Dr. Wai-
_ lis and others, and their Properties are now
common enough ; fo that there’s fcarce. any
thing new left to be difcover’d concerning
them. But the famous M. De Lattire in a
late Treatife, having fhewn fome of the Pro-
perties of the Primary Epicycloid, the moft
‘Ingenious Mr. Ca/mell did upon that occafion
not: only demonftrate that the Menfuration ~
of the whole Epicycloidal Space, obtain’d al-
-fo in the parts of the faine, but alfo gave
a perfec Quadrature of the Curvilineal Space
UPL. But while I was enquiring after the
Demonftration of this Quadrature, which is
“pot very obvious, nor as yet given by the In-
yentour, I light upon the following general
Pro-
ed x
178 Mifcellanea Curwfa.
Propofition, by the help of which all forts of
Curvilineal Spaces, as well of the Cycloidal
as Epicycloidal kind, as well the whole
Spaces as the parts, are meafur’d. And far-
ther, not only the Spaces VPL, but alfo in-
numerable others, QTP and VQOTL, are de- —
monftrated to be capable of an exact Qua- .
drature; and this not only in the Pramary
Epicycloids, but but alfoin the Prolare and
contracted ones. By kes
The Prupofition is as follows.
Propofition. ,
The Area of any Cycloid or Epicycloid,
either Primary, Prolate, or Contraéted, is
to the Area of the Generating Circle, and —
alfo the Areas of .the generated parts in
thofe Curves, are to the Areas of the Ana-
logous Segments of the Circle; as the Sum
of twice the Velocity of the Center, and —
the Velocity of the Circular Motion, to the
Velocity of the Circular Motion. ,
Demonfiration, * —
be (Fig. 32.)
Let YPQRSUB be any Epicycloid defcrib’d
by the Revolution of the Circle ULB, up-
on the Circular Bafis YMINB. Let the Cen-.
ter of the Generating Circle be in ¢, and
drawing cMK, let the Circle ftand upon the —
Bafis in the point M, and let the defcribing —
point be S. Now diftinguifhing the Motions,
Jet the point S firit of all be carried by the —
Circular Motion into R, fo that the Arch
| MifceHanea Curtofa. 17979 -
SM is increafed by the indivifible Particle |
RS. Next fuppofe the Center c to be tranf-
ferr’d to C; by which Motion the Segment.
RSM being brought into the Pofition QTN,
_the point Q will touch the Curve. ’Tis
plain that the Triangle RSM is the Mo-
mentum or Fluxion of the Segment of the
Circle, and that the Trapezium QSMN is
the Fluxion of the Curvilineal Space gene-
rated in the fame time. And fince SM,
RM, QM, are fuppos’d to differ but by a
pont from one another, let the little Area
QSMN be conceiv’d to confift of the three
Sectors RMS, RMQ, MQN; and fo the
little Area RMS to be to the little Area
_QSMIN, as the Angle RSM to the Sum of the .
_ three Angles RMS-|-RMQ-|-MQN. But the
AnglesRMQ-|- MON, are equal to the Angles
MCN -- MEN, or to the Angle cMC ; be-
-caufe of the Lines RM, QN, inclin’d to one |
egret ‘in an Angle equal to MKN, and
becanfe of the Angle MQN equal to ] MCN
(by Excl. 3. 20.) confequently the Angle
RMS is to the Angles RMS-|+-cMC, that
As (by the fame Propofition mention’d) the
Arch 3 RS to the two Arches Cc-+-3 RS, or |
RS to 2Cc IRS, as the little Area RSM,
‘to the little Area QSMN, or as the Mo-
mentum of the Circular Segment QTN, to
the Momentum of the Epicycloidal Segment
OSYMN generated in the fame time. And
ce thefe Momenta are ever in that fame
Ratio, where-ever the point Q_ be taken, ’tis
Manifeft that the Areas QTN, QSMYN
themfelves, generated from thefe Momenta,
Thave alfo the fame conftant Ratio, viz. -
oc N 2 tne
i180 = Mazfcellanea Curiofa.
the Velocity of the Circular Motion RS, to
double the Velocity of the Center, adding
the Circular Motion, or 2Cce-| RS: As al-
fo the Area UBZ to the Area QUBN, and
confequently the Semicircle ULB to the Cur-
vilineal Space UQYNB.. Wherefore the
Propofition is manifeft. | |
And there is no other difference in the
manner of demonftrating, if the generating
Circle moves upon the Concave fide. of the
Arch, except only that the Angle cMC, in
this cafe, is the difference of the Angles.
MCN, MEN. But if the Bafis were a right
Line, then MKN vanifhing, and RM, QN,
being parallel, the Conftrudion will be ea-
fier. 1 forbear drawing Corollaries from this
Propofition, fince they are obvious. But
now in all thefe Curves, the Portions that
are Analogous to thofe Portions which Do-
ctor Wallis has found capable of a perfect
Quadrature in the Primary Cycloid, are here
alfo equally fquarable; which eafily follows
from what has been faid.
Upon the Center K, thro’ the point Q.,
draw the Circular Arch QZ, and draw ZB
cutting off the Segment ZLB = the Segment
- QTN.. Then bifecét the Semicircle UB in L,
and thro’ the point L and on the Center K,
defcribe the Arch PL cutting the Epicycloid
in P, the generating Circle in T, and the
Chords QN, ZB, in y and X. Let the Arch
VZ.= a, its Sine =, the Radius of the
generating Circle = r, the Radius of the
-Bafe=R, and the Arch CE or the Motion
of the Center =m. It is plain that the
Sector CKE, is to the Space XyINB, as the
| | © ~~ Square
——, Mifeeltanea Curtofa. 18%
Square of KE, to the difference of the
- ‘Squares of KL and KB, or asRR-l 2Re-brr,
fae a 201, chat is, as R.F t,.to 2r, or
KE to BV. And confequently the Rectan-
gle BE. x CE or rm is equal to the Space
XyNB. But the Space VZB is equal to the
‘Rectangle 3 ar 13 sr, and. fo according to
our Propofition it will be as 4 to 2m, fo } ar
ee May is, 7 ay
“S sr, to SL ae equal to the Curvilineal
Space QUZLBNQ: From hence fubftra&
the Space XyNB==rm, and there remains
the. Space QUZXy ==. And fince the
‘Spaces ZXL, QyT, are equal, the Space
QULTQ. fhall alfo be equal to ae ed ficre-
fore when ato m, or the Circular Motion
is to the Progreflive Motion of the Center,
‘in a given Ratio, there will be a perfec
Quadrature of the Curvilineal Spaces
~QULTQ. And the whole Space UPL, will
_ be to the Square of the Radius BE, in the
fame Ratio (m toa) of the Motions, that
is in-every Primary Epicyclotd;-in-the Pro-
portion of the Radii, KE, KB, which is
Mr. Cafwell’s Propofition.
But the leffler Spaces QULTQ will be to
- one another, as the Sines of the Arches UZ;
and the Triangular Spaces QTP, by the fame
Argument, will be as the verfed Sines of the
‘Arches QT or ZL, and confequently are al-
fo fquar'd. After the fame manner it will
be prov’d, that the Spaces par, pLu, par,
are ever to the Square of the Radius BE
N 3 mete
782 0 )©=— Mifcellanea Cueiie one
Cin all thefe Figures) in the aforefaid Ratio
of m toa; and their Portions pqt, as the
verfed Sines of the intercepted Arches qt;
but the remaining Segments as qtYA,. qtTa, &c.
will be as the right Sines of the Compli-
ments of the fame Arches 5 ag
But the Ratio of the Velocities, m tO 4, is
compounded of the Ratio of the Radii KB, :
BE, and the Ratio of the Angles ~ CKE,
VEZ, equably defcrib’d together; and con- -
fequently giving that Ratio of the Angles,
all the foremention’d Epicycloidal Spaces
will be fquar’d alfo. :
I can eafily draw Tangents to all thefe
Curves, as alfo I feem to my felf to have
gotten their Reifications, from fome Areas
Analogous to. them ; which may give occa-
fion_ to a more particular handling of as
Family of Curves another time. | :
Mifcellanea Curiofa. 883
A Method of R.aifing an infinite
Multinomial to any given Power,
or Extracting any given Root of
the fame. fe ES
_. By Mr. A. De Moivre:
_ 9F PIS about two Years fince, that confi-
-°§ dering Mr. Newtons Theorem for
Railing a Binomial to any given Power, or
Extracting any Root of the fame; I enquir’d,
whether what he had done: for a Binomial,
could not be done for an infinite Multino-
mial. I foon found the thing was poffible,
_and effected it, as.you may fee in the follow-
ing Paper; I defign ina little time to fhew
the Ufes it may be applied to: In the mean
while, thofe that are already vers’d in the —
Doérine of Infinite Series, and have feen
what Applications Mr. Newton has made of
his, Theorem,.may of themfelves derive fe-
veral Ufes from this. :
I fappofe that the Infinite Number Multi-
Momial sig 1 bez \ cz? | dz* en”, ec. m
is the Index of the Power, to which ‘this
Maltinomial ought to’ be Rais’d, or if you
Will, "tis the Index of ‘thé Root which is to
be Bxtiadedy I fay that this Power or Root
of'the Multifomial,\ is fach a Series as |
Bh e@iarcieit iss yd obi buss ye vig
“Bhieao! N 4. For
184 Mifcellanea Curiofa.
For the underftanding of it, it is only ne-
ceflary to confider all the Terms by which
the famé Power of z is multiplied, 1m or-
der thereto I diftinguifh two things in each —
of thefe Terms; Firft, The Product of cer-
‘tain Powers of the Quantities, 4, 6, ¢, d, Xc-
Secondly, The Uncia (as Oughtred calls em)
prefixt to thefe Produéts. To find’ all the
- Produéts belonging to the fame Power of z,
to that Produét, for inftance, whofe Index ts
mr (where ry may denote any integer Num-
ber) I divide thefe Products into feveral Claf-
fes; thofe which immediately after fome
certain Power of 2 (by which all thefe Pro- —
du€&ts begin)! have 6, I call Produéts of the
fir Claffis; Bor Example, a™—~* be is a Pro-
du@ of the fir Cla/fis, becaufe 6 immediate-
ly follows 2”—*3 thofe which immediately
after fome Power of a have c, I call Products —
of the fecond Cla/fix, fo a2”? ced is. a Produc ~
of the fecond Claffis, thofe which immedi-
ately after fome Power of a.have 4, | call
Enos of the third Cla/fis, and fo of the
reit. 7 | AES Ween
This being done, 1 multiplyall the Products
\Wa
by @ all the Terms, be+
Es ae
Mifcellanea Curiofa. ¥8g
longing to z mrr—4_ except thofe of the firft,
fecond, and third Claffis, and fo on, till i
meet twice with the fame Term. Laftly, I
add to all thefe Terms the Product of a”—"
into the Letter whofe Exponent is r--1
“Here I muft take notice that by the Ex-
ponent of a Letter, I mean the Number
which expreffes what Place the Letter has in
the Alphabet, fo three is the Exponent of
fie etter: ¢ * becaute.' the Cetter. ¢ is the
third in the Alphabet.
‘At is evident that by this Rule. you may
ait find all the Produéts belonging to the
feveral Powers of z, if you have but the Pro-
dud belonging to 2” viz. a™
To find the Uzcie which ought to be pre-
fixt. to every Product, 1 confider. the Sum of
Units contain’d in the Indices of the Let-
ters which compofe it (the Index of a ex~
cepted) I write as many Terms of the Series
mrm—-1x%*m—2x m—3,&c. as there are
Units in the Sum of thefe Indices, this Series
is to be the Numerator of a Fraction, whofe
Denominator is the Produé of the. feveral Se-
ries TR2% 3X4 % 55,06 1X 243 e.4X.5., oc,
TX2KX3%4%5X%6, &e. the firf€ of which
contains as many Terms as there are Uvits in
the Index of b, the fecond as many as there
are Units in the Index ‘of c, the third as ma-
“hy as there are Vuiets in the Index of d, the
fourth as many as there are Units in the Index
of e, &e. eye é
De.
&
©.
FS
a
186 Mifcellanea Curiofa. —
Demonftration.
To raife the Series az -|-bzz--ez3 + dz*,&e.
to any Power whatfoever, write fo many Se-
ries equal to it as there are Umits in the In-
dex of the Power demanded. Now it Is evi-
dent that when thefe Series are fo multipli-
ed, there are feveral Products in which there.
is the fame Power of z, thus if. the Series
ez tbe 4 cz). tdz*, | &c.. iss raed 4048s
Cube, you have the Produds 6°z°%, abez®,
aadz®, in which you find the fame Power z°
Therefore let us confider what 1s the Condi-
tion that can make fome Prodads to contain
rhé fame Power of z, the firft thing that will
appear in relation to it, is that in any Pro-
duct whatfoever, the Index of z is the Sum
of the particular Indices of z in the multi-
- plying Terms (this follows from the Nature
of Indices) thus 632° is the Produ@ of bz?,
b27, bz?, andthe Sum of the Indices in the
multiplying Terms, is 2-2‘ + 2&6 sabes
in the Prodad& of az, bzz, cz3, and the Sum
of ‘them “Indices of z‘in the multiplying
Terms is 174-2136 aadz® is the Produd&
of az, az, dz*, and the Sum of the Indices
of 2 in'the multiplying Terms is 1-+it4=6;
the’ next thing’ that appears is, thatthe In-
dex of x in the multiplying Terms is the
“fame with the’ Exponeiit © oF the Letter to —
to which z is join’d, from which two Confi-
derations it follows, that, To have all the Pro-
ducts belonging to a certain Power of 7, you muff
find all the Produtts where the Suna of the Ex-
HE ; _ pouents —
-
Mifcellanea Curiofa. 1 87
ponents of the Letters which compofe ’em fhall
always be the fame with the Index of that Power.
Now this is the Method I ufé to find eafily
all the Products belonging to the fame Pow~
er of z, Let m-|-r be the Index of that
Power, I confider that the Sum of thé Ex-
ponents of the Letters which compofe thefe
Produéts muft exceed by one thofe which be-
long to m™1’—", now becaufe the Excefs of
the Exponent of the Letter b above the Ex-
ponent of the Letter 4, is one, it follows
that if each of. the Produéts belonging. to
z\7—! is multiplied by 6, and divided by 4,
you will have Products the Sum of whofe Ex-
ponents will be #-|-r;, Likewife the Sum of
the Exponents of the Letters which com-
pofe the Products belonging to z™I™ exceeds
‘by two the Sum of the Exponents of the
Letters which compofe the Products belong-
ing to z™!’—?; Now becaufe the Exponent
of the Letter a is lefs by two than the Ex-
ponent of the Letter c, it follows, that if
each Produét belonging to zmtt—2 is multic
plied by ¢ and divided by 4, you will have
other Products, the Sum of whofe Exponents
is till m-lr; Now if all the Products be-
longing to zit1—2 were multiplied by ¢ and
~ divided by 2, you would have fome Produds.
that would be the fame as fome of thofe
found before, therefore you muft except out
ef em thofe that I have’ call’d Produ&s of
the firft Claffis; what I have faid fhows why ©
all the Products belonging to hig Sd so
cept thofe of the firft and fecond Cla/fis pa
188 Mifcellanen Curiafa. |
~
Letters of each Solid may ‘be changed.
be multiplied by d and divided by 4:, Laftly
you fee the Reafon why to all thefe Products
is added the Produ@ of a”—" by. the Let-
ter, whofe Exponent. is rl 14 "Tis becaufe
the Sum of the Exponents is ftilla-r. ©
As for what relates to the Unciaz;, obferve
that when you multiply 2z-bzz }-cez3 4-dz*,
&c. by ax bzz ez} -- dz*, &c. each Let-
ter: 4)°b,) co; d> &e. ofthe Tetone semes i
multiplied by each of the Letters 2, 6, c,d,
&c. of the firft Series; Thus the Letter 2 of
_ the fecond Series is multiplied by the Letter
& of the firft, and the Letter_d of: the: fe-
cond Series is multiplied by the Letter a of
the firft; therefore you ‘have the: two Planes,
ab, ab or 2ab for the fame-reafon you have
2ac, 2ad, &c.,. Yherefore you muft prefix to
each Plane of thofe that, compofe the Square
of the infinite Series az-\ bez-\ez3, &e,
the Number which exprefles\how many ways
the Letters of each Plane may be. changed;
likewife if you multiply the Produd of the
two preceeding Seriés by az-bzz*}ex3, &c.
each Lettera; 6; ¢, dj ofthe third ‘Series is
multiplied by each of the Planes form’d by
the Produét of the® firft and fecond Series;
Thus the Letterzis multiplied by the Planes -
bc and.ob;.the,Letter @ is multiplied, bya
and ca; the Letter.¢,is multiplied by. and
ba, therefore yous have, the fix. Solids;, abe;
ath, bat, bea, cub, cha, or Sabe ;:, Therefore
you, muft» prefix. to;cach Solid whereof the
Cube of the infinite Series :is compos’d, the
Number which expreffes haw. many ways aps
| FAs
, genes ~
| Mifcellanea Curiofa. 4 89
generally, You muft prefix to any Produ where-
of any Power of the infinite Series az bzz--
cz>, &c. w composed the Number which expref-
_fes how many ways the Letters of each Produ&
may be changed. |
Now to find how many ways the Letters
of any Product, for inftances a”—” bc? at
may be changed; this is the Rule which is
commonly given: Write as many Terms of
the Series 1% 2%3% 4x95, &c. as there are
Units in the Sum ofthe Indices, wz. m-2
th-p-r, let this Series be the Numera-
tor of-a Fraction whofe Denominator fhall
be the Produét of the Series 1 x 2x 3% 4% §,
ee O28 2h 4X54 Ce. 1X 2X 3X'4% 8X6, *
&c. 1X 2% 3x%4x 5, &c. whereof the firft is
to contain as many Terms, as there are
Units. in the firft Index m—x; the fecond
as many as there are Units in the fecond In-
~ dex 43 the third as many as there are Units
in the third Index p;_ the fourth as many as
there are Units in the fourth Indexs;y. But
the Numerator and Denominator of this
Fraction have a common Divifor, wz. the
Series 1x 2%3x%4%x5, &c. continued to fo
many Terms as there are Units in the firft
Index m—2; therefore let both this Nume-
rator and Denominator be divided by this _
common Divifor, then this new Numerator
will begin with m~x#-\1, whereas t’other
began with 1, and will contain fo many Terms
as there are Units in b-+p-\r, that is, fo
many as there are Units in the Sum of all
the Indices, excepting the firft; as for the
new Denominator, it will be the slurs of
. three
_
190 =6§ Mifcellanea Curtofa.
three Series only, that is, of fo many as their
Indices, excepting the firft. But if it hap-
pens withal, that 2 be equal to h-t-p-r as
it always happens in our Theorem, then the
Numerator beginning by m—x-\.1, and be-
ing continued.to fo many Terms as there are
Units in b‘\ p-\r or 2, the laft Term will be
m neceflarily, fo if you invert the Series and.
make that the firft Term which was the laft, ©
the Numerator will be mxm—txm—z2
x m—3, &c. continued to fo many Terms
as there are Units in the Sum of the Indices
of cach Produét, exceping the firft-Index.
There remains but one thing to demontftrate,
which is, that, what I have faid of Powers
whofe Index is an Integer, may be adapted
to Roots, or Powers whofe Index is a Fra-
ction; but it appears at firft fight why it.
fhould be fo: For, the fame Reafon which
makes me confider Roots under the Notion
of Powers, will make me conclude, that what-
ever is faid of one may be faid of tother ;
however, I think fometime to give a more
formal Demonftration of it. . ised
SS sarees acy See
LTRCOTEM.
a Sd
ae+hz? +c27 + dz*+ezi+fz2° + 9Z7 + Az? + 429 KC.
Page Lg 27
72
a
72
Z
Me jy TL M2» WL 7 W—-2
pm n”’~d
w
+z x 22 y Fe? « -3 a Meat al
+ fix Minty mma QQ” 3 Pd
MAR 2 2
+Brmee ac + Be ttn tee 29 be?
BES w— Z
age e é +3 xB gag”? be
eg ea a Ce
m—L
+poan f
t
( = 7 22 od
AP x Meaty Peay Me 1 Mey Bian 8 pp te MER hee
e
+ ¢ Mik y WB y 3 x mt = he
+ Fx Mipty moag mes a™—* Yd
x Mga y Many med g™-# fica
+7
PRX Baty m2 Qe? be
Hee, wea 2 hed
RR ARE
= 2 2 2 m3 m+
4 ma MAG gt 4 Bey reg gfe MAA 4 my moty 2g 3 hig 3
PP iar tpt, Baa" "be" 4 my mes mary mee ghee tS
+H x Bbptx Manz 2"? HM ¢
se N
j 2
oh ae
4
ty eal
So tes Seiichi Lovanbod a bl
ai i a is mca:
Ho ee aa
Py teotnnhe y= nah aot iin
: ’
j.
A ed
1 oe
pene, §
§
. ‘ Lig encima time a
ek yi - ¥
i Sere
Oe) oe plete tn eosin Sacs ss os Sey lta PONT
Fe j
Wid ,
?
= doce
siewaaign
aah.
CUM
RR Hea ACN
Maifcellanea Curiofa. 191 |
A Method if eee a the Root
mi an Infinite Equation.
By A. De Moivre, ERS.
THEOREM.
Pu arbarperstdet fort f°, &e. = gy!
“thyyrry? ee Ai -+my®, &c. then will
z be o£ fy
ir
“
a
io) ul = cA}
ay?
ied
ke bBB—2bAC— 3¢ AAB = dAt
eee , ——y*
a
1—2bBC—2bAD—3 c ABB—3¢ AAC—sd A> B
—eA?
OEE ee ys
i
Bt morr
m =--2 BD --- bCC---2)_AE --- cB} --- 6c. ABC
won 3¢- AAD == 6A AABB---41 A} C---52.A*B
menhite
ena meena einer
y°, &e.
a
For the underftanding of this Series, and oe ‘
“in order to continue it as far as we pleafe,
it is to be obferv’d, 1. That every Capital
p Letter: is equal to the Coefficient of each
pre-
192 Mifcelanea Curiofa..
preceding Term ; thus the Letter B is equal to
the Coefficient h—bA A
2. That the De-
nominator of each Coefficient is always a.
3. That the firlt. Member of each Numera-
tor, is always a Coefficient of the Series
gy--hyy-iy?, &e. viz. the firft Numerator
begins with the firft Coefficient g, the fecond —
Numerator with the fecond Coefficient 4;
aud fo on. 4 That in every Member after
the firft, the Sum of the Exponents of the
Capital Letters, is always equal to the
Index of the Power to which this Member
belongs: Thus confidering the Coefficient ~
pulser ee —— bes 4 9
pic laleDRcirano NE cee ; which be-
longs to the Power y*, we fhall fee that in
every Member bBB, 2b:AC, 3CAAB, 4A*,.
the Sum of the Exponents of the Capital.
Letters is 4, (where I muft take notice, tliat”
by the Exponent of a Letter, I mean the
Number which expreffes what Place it has in
the Alphabet ; thus 4 is the Exponent: of the
Letter D) hence I derive this Rule for find-.
_ ing the Capital Letters of all the Members
that belong to-any Power 5 Combine the Capi-—
tal Letters as often as you can-make the Sum ‘of
their Exponents equal to the Index of the Power to
which they-belong. §. That-the Exponents of |
the fmall Letters, which are written before
the Capitals, exprefs how many Capitals
there is in each Member. 6. That the Nu-~
merical Figures or Uvese that occur in thefe »
Members, exprefs the Number of -Permuta-.
159 TIONS
‘
, Mifceilanea Curiofa. . 192
tions which the Capital Letters of every
_ Member are capable of. | |
_ For the Demonftration of this; fuppofe
= Ay- Byy-|-Cy3-|-Dy*, &c. Subftitute this
‘Series in the room of z, and the Powers of
_ this Series, in the room of the Powers of <3
there will arife a new Series, then take the
Coefficients which belong to the feveral
Powers of y, in this new Series, and make
them equal to the correfponding Coefficients
of the Series gy-|-hyy-|-zy3, &c. and the Co- |
efficients 4, B,C, D, &c. will be found fuch -
vas I have determin’d them.
But if any one defires to be fatisfied, that
the Law by which the Coefficients are form’d,
will always hold, P’ll defire em to have re-
courfe to the Theorem I have given for. rai- —
fing an infinite Series to any Power, or ex-
tracting any Root of the fame; for if they
ee ufe of it, for taking fucceflively the
Powers of <Ay-\-Byy-|-Cy?, &c. they will fee ©
‘that it muft of neceflity be fo. I might have
made the Theorem I give here, much more
‘General than it is; for I might have fuppos’d,
abe) ge a ai [2 res f opm oy by 3 -|- ails
‘&c. then all the Powers of the Series .4y-|-
Byy-\-Cy? ,&c. defign’d by the univerfal Indices,
‘Mmuft have been taken. fucceflively ; but thofe
who will pleafe to try this, may eafily do it,
Dy means of the Theorem for raifing an infinite
Series to. any Power, &c.
\This Theorem may be applied to what is
called the Reverfion of Series, fuch as find-
Ing the Number from.its Logarithm given;
the Sine from the Arc; the Ordinate of an
. en
Rage 3 A 18a toe c ao Rua tS ERMe eit
ata Aes
194 Mifcellanea Curwfa. :
Ellipfe from an Area given to be cut from |
any Point in the Axis: But to make a pat- —
ticular Application of it, Pil fappofe we have —
this Problem to folve; vz. The Chord of
an Arc being given, to find the Chord of ©
another Arc, that fhall be to the firft as
“ to 1. Let y be the Chord given, % the
Chord required ; now the Arc belonging, to—
cos he
we Chontyy is) ¥ ++ 6dd a 4oa* is 112a°—
&c. and the Arc belonging to the Chord hi
eos i oge) an SBE arft of
a -|- sod! BPS Flic the firft of
thefe Arcs is to the fecond as_1 to 73
therefore multiplying the Extreams and
Means together, we fhall have this Equa-
tion: : ic
12) 3) ee
64a god" ae one be, be 7) odd
Sy Bam tit ISIE ek
! godt 11246 mee 3 ; er
Gompare thefe two Series with the two
Series of the Theorem, and-you will find”
is z -|-
R=
: Nea
| eat, b=o, om Fidos = oa Ia
32 x
x rr n : |
Bcc. got, b=0, 1 Gd =O I godt
4 Yr 3
m—o. &c. hence z will be =ay-4- cae
: ‘ At eaig Td
Re tr ny -\- bees pte: tiee |
| 2 4 3das ? Ge ty.“ as
to
| MifceHanea Curiofa. 195
to denote the whole preceding Term, which
will be the fame Series as Mr. Newton has
firft found. - Hie |
By the fame Method, this general Pro-
blem may be folv’d; the Abfciffe correfpond-
ing to a certain Area in any Curve being gi-
ven, to find the Abicifle, whofe correfpond-
ing Area fhall be to the firft in a given Ra- ©
- to. .
The Logarithmick Series might alfo be
found without borrowing any other Idea,
than that Logarithms are the Indices of
Powers: Let the Number, whofe Logarithm
we inguire, be 1-|-z, fuppofe its Log. to be
az--ozz-|-cz3, &c. Let there be another
“Number i-|-y; thereof its Logarithm will
be, ay-|- byy-|-cy?, &e. Now if 1--z=
1-/- yl, it follows, that az -|- bzz-|-¢z3, &c.
ay -|-byy-|-cy?, &c. :: 2, 1. that is, az-—-
zz --0z3, &c. = nay -|- nbyy -|- my? &c.
Therefore we may find a Value of z-expreft
by the Powers of y; again, fince 1-+<=
a-ky' therefore z = 1 = aa 1, that is,
Sa Sy. a ate Soa, iS
ae bom IY I 2, 3
y*®, &c. Therefore z is doubly expreft by
the Powers of y. Compare thefe two Values
‘together, and the Coefficients 4, 6, ¢, &c.
will be determin’d, except the firft 2 which
“may be taken at pleafure, and gives accord-
ingly, all the different Species of Logarithms. .
9:2. . An
te RS ere ee ce: OPN
MEE Ay Er
- Te ‘
Must a
196 ©Mifcellanea Curiofa.
An Experiment of the Refraction
of the Atr made at the Com-
mand of the Royal Society,
March 28.5699... 4
By J. Lowthorp, 4, M.
E took’ a~ Cylinder -of Caft-Brafs
3 (Fiz. 33.) ABCD, and cut one end of
it CD perpendicular ‘to the Axis ax, the
other end AB enclin’d to it at an Angle of
about 27°. 30. and therefore the Perpendi-
cular to this enclining plain, pe, and the Axis
of the Cylinder- ax comprehended an Angle
pea of about 620. 30d. Thefe ends were
groun’d very true upon.a Glafs-Grinder’s
Brafs-Tool, and each of them ‘was compatft
_about with a narrow Ferule of thin Brafs bb00.
Into the upper fide of the Cylinder at EB was
“folder’d the Brafs-Pipe EF} and into the un-
der fide at 6 the other Brafs-Pipe GH; the
former of thefe Pipes being about 3 Inches
long, and the latter. 6 Inches. Upon the
Plate ddd were fixt to two other Plates LL
perpendicular to it and parallel to each other.
Bach of thefe two Plates had an: Arch of a
Circle € equal to the.Circumference of the
Cylinder) cut out of its upper Edge, fo that
when the Pipe GH was let :through ‘a-hole
‘near. -the-middle of the: Plate ddd,:the Cy-
linder fell into the Arches; and being fa-
. ften’d there with Soder, the Axis ax laid pa-
~kallel to the Plate.ddd and about an Inch ie
| . sgt ie . ieee 2 ha 8
UMifcellarea Cupafa. | 197°
half above it. The perpendicular End of -
“the Cylinder DC was clos’d with an Objeé
Glafs of.a feventy fixth Foot Telefcope oo 3,
and the. other end AB, with a vwell polith’d
flat Glafs ff; which was carefully chofen to
tranfit the Obj-& diftina enough notwith-
ftanding its Obliquity to the Vifnal Rays.
-The Ferules were well fild with Cement
round about the Edges of the Glafs, and they
Jaid flat and every where touch’d the fmooth
Ends of the Cylinder, that they might firm-
ly refift the preffure of the excluded Air.
Inftead of a Ciftern (as in the Torricellian
Experiment) we made ufe of the Inverted
Siphon of Brafs (Fig. 34.) MNO, foder’d to
the Plate gzg. One of the fides "MN ftood
perpendicular to the Plate, and the other
fide NO enclin’d to it, and was fupported.
‘near the upper end O with a little prop &k.
- Wethen placed the Cylinder (as in Fig.33.)
upon a Table which was well faiten’ d to a
firm Floor; the Pipe GH was let through a |
hole, and the Axis laid almoft parallel to the
fides of the Table, and the Plate ddd was
nail’d. down to it. The Tube of the Tele-
feope ff, with the Eye-glafs, was apply’d to
the Object Glafs, and a Hair fix’d within
it at the common Focus of both Glaffes in the.
Axis of the Cylinder. continw’d, x. . Upon the
Floor (under the Cylinder). we id the
Plate ggg with the Inverted Siphon upon it,
-and join’d M to H by the InfeSion of the
Glafs Tube T. The Joints were very care-
fully. clos’d with:Cement: And then they
awere coyer’d over with pieces of a Bladder
B - wee hard with ftrong Thread. ‘There
; “QO 3 was
ni qe aa iy Av.
WT Oy pea eae a MaRS Ter Nee
$ > 4 Sobel Sine igi eS
198 MifceHanea Curwofa.
was alfo a Bladder ty’d below each Joint at’
m, and when it was filld with Water it was
tyd above it at 7; fo that no Air could
come to the Cement, or infinuate it felf
through its Pores or Siffures if any happen’d
to be left unclos’d. pM
It is not (1 think) an unneceflary trouble,
that in this account of the Apparatus I have
mention’d fo many minute Circumftances,
for we found it difficult enough to exclude
the Air, and almoft impoffible to difco-
ver the very little holes through which fo
fubtil a Fluid would freely enter and poflefs
the Spaces deferted by the fubfiding Mer-
-cury. But with all this Precaution the Ex-
- periment fucceeded at laft, as 1 wifhd, af
ter this manner. Ee ‘ :
We plac’d the Obje& 2 (which was a black
Thread fliding in a little Frame over a piece
of white Paper) in the Axis of the Cylinder
cx continu’d to it, we filled the Pipes and Cy-
linder with Mercury; and having ftopt the up-
permoft Pipe at F with the little lron-{topple
K and clos’d it at the other Joints, welet the
Mercury run out gently at O into the Blad-
der v, till it remain’d fofpended at the ufual
height (as in the Barometre) leaving the
fpace above it between the Glafles 00 and ff
void of Air. We then found the Object,
which before appear’d in the Axis at x, raisd
confiderably above it; and we reduc’d it to
appear at x by removing it from @ to x.
The Axis therefore, of the vifual Ray +a,
(which was alfo the Axis of the Cylinder) «a,
falling- perpendicularly on the void fpace in
the Cylinder paft through it without any Re-
— —— Mifcellanea Curifa. 199
fra&tion: But emerging obliquely into the
‘Air, it was refra@ted towards the Perpendi-
cular pe, and there receiv’d a new Direction
tox. And therefore the fpace ax fubftend-
ed the Angle of Refraction acx; which we
meafur’d and found as follows. .
- The height of the Obje&t- Inches Depths
above the Axis of ye
fual-Ray 2x the unre- ° 425
‘ fracted bp ey
The Diftance of the Ob-
: je from the Refract- 6
ing Plain, @c. about Ca
oeepeeect Or
Therefore the Angle of , i
-Refra@tion acx was — ¢ Cee Fae
The Angle of Emerfion oy :
_ pea (by the Conftrudti-
on of the Cylinder) a”
was oe
Therefore the Angle of)
Incidence pex== (=peap 62. 27- coe
ate Od | |
And therefore univerfally (according to
the known Laws of RefraGtion) |
The Sines of the Aiges & eesti
of Incidence being — 3.
The Sines of the paglc f bbs
Emerfion are — — oe
And the Refractive Pow-2 ss _-3
er of the Denfe Air —-$ ;
By the Refractive Power of a pellucid Bo-
dy, limean that property in it whereby the
Oblique Rays of Light are diverted from
“their dire& Courfe ; and which is meafur’d
Ne ‘ O 4 DY
*.
i?
200. Mifcellanea Curiofa. ~~
by the proportional Differences always ob-—
ferv’d between the Sines of the Angles of In-_
cidence and Emerfion. ngchs es.
This Property is not always propor-—
ie cb Nites aie tt
me
cor
tional to. the Denfity (at leaf not to the
Gravity) of the Refra@ing Medium. For
the Refra@ive Power of Glafs to that”
of Water, is as 55 to 34, whereas its Gra-_
vity, is as 87 to 34; that is, the Squares of
their Refractive Powers are (very meat) as
their refpective Gravities. And there are
fome Fluids which though lighter than Wa-.
ter yet have a greater power of Refraction ;
thus the Refractive Power of Spirit of Wine
(according to Dr. Hoek’s Experiments, AZ-
crog. p. 220.) is to that of Water, as 36 to
33, and its‘Gravity reciprocally, as 33.to
36, or 362. But the Refra&ive Powers of
Air and Water feem to obferve the fimple
Proportion of their Gravities, dire@ly;.as I
have compar’d them in. the following Table.
The Numbers there exprefling the Refra-
ction of Water are taken from the Mean. of
-* Nine Obfervations at fo many feveral An-
gles of Incidence, made Jar. 25. 1647. by
Mr. Gafcoigue the ingeninos Firft Inventor of
the Micrometer, and the ways of meafuring
Angles by Telefcopes, and thofe of Air are
produc’d by; the Experinient above related.
* Iam indebted. for them to Mr. Flamfteet, who. had
cover'd them with his-Objervations, ‘andi feveral Paffages
relating to them,. from.his, Letters to-Mr. Crabtree, which
mere pappily prefery'd in the Time of our Civil War by.
Me
Sir Jonas Moor, ‘and Mr. Chriftopher Towneley ; and
ere now in- the’ Hands ‘of Mt. Richard ‘Towneley of
hg in Lancathire,: by-whow rhey were inparted to
fie ay age oi ems
Ts i ane:
ts,
gS peat
Nine te
eee Mite ye
Daitat ad sete Me ete th Ay
rr
P i
a
f
7
: nis '
: 4
F i ‘
j
, w
i ")
Ni
Hi
‘
i
,
5 +
igure
Ph
F
“ri 4 et 3
: 7
D °
zag
vy
»
Vass.
I
wi
i
;
ERA We
vals Bg
| Mifcellanea Cuvofa. 201
The (aflum’d) Sines of 1 Water. Air. _
the Angles of Inci- p150000: 100000
dence through — J
- The Sines of the corre-} |
fpondent Angles of ?134400. 100036
_Emerfion out of — J hae:
eS pial ae $AA00 70. 85 36
The Specifick Gravity)
{Gf as 900 to 1 at the | :
time of the Experi-.>34400.... § 38 ©
mént) of or (if as| gO:
850 to 1) of —-——+-— J Y
‘From hence it feems very probable that
‘their Refpedive Denfities and Refractive
Powers are in a juft Simple Proportion: And
if this fhould be confirm’d by fucceeding Ex-
periments, made at different Angles of Inci-
dence, and with Cylinders continuing ex-
haufted through feveral Changes of the Air,
it would be more than probable that the Re-
fraétive Powers of the Atmofphere are every
-where, at all heights above the Earth, in pro- .
portion to its Denfities and Expanfions. And-
here it would be no difficult matter to trace -
the Light through it, thereby to terminate
the Shadow of the Barth; and (together with
proper Expedients for meafuring the Quantity
of Light illuminating an Opaque Body) to ex-
amine at what diftances the Moon muft be
from the Earth to fuffer Eclipfes of the ob-
ferv’d Duration. . This Limitation is confide-.
able enough in Aftronomy, abundantly to
‘recompenfe the Trouble of profecuting fuch ©
anew Experiment. =.
ee sao ae AA
202 Mie tes Cores. |
A Dim concerning a Mabal
of Difcovering the true ‘Moment
if the Sun’s Ingrefs into the Tro-
pical Sines. :
es E. Halley.
T may perhaps pafs ee a Paradox, if bot
feem extravagant, if I fhould affert that
it is an eafier matter to be affur’d of the
Moments of the Tropicks, or of the Times
of the Sun’s Entrance into Cancer and Capri~
corn, than it is to-obferve the true Times
of the Equinodtials or Ingrefs into Aries and
Libra. know the Opinion both of Ancient
and Modern Aftronomers to the contrary ;
Ptolemy fays exprefly, Tas trav resrav tupaces
~ Such venetrss ivan s And Ricciolus begins. his
Chapter of the: Solftitial Obfervations with
_thefe words, Mer:to Snellius, iz notis ad ob-
fervationes Hafliacas, pronunciavit, Her culei effe
laboris vitare in Solttitiis obfervandis errorem
quadrantis diei, and this becaufe of the ex-
ceeding flownefs of the change of the Sun’s
Declination on the day of, the Tropick, be-
ing not a quarter of a Minute in twenty
four Hours. ‘This indeed would make it ve-
sy difficult, nor would any Inftruments fuffice
to do it, were the Moment of the Tropick
_fo-be determin’d from one fingle Obferva-
tion. But as three fubfequent Biediieone
made
Mifcellanea Curiofa. 202
‘made near the Tropick, at proper Intervals
of Time, I hereby defign to fhew a Method
to find the Moment of the Tropicks capable
- of all the ExaGnefs the moft Accurate can
defire; and that without any confideration
of the Parallax of the Sun, of the Refradi-
ons of the Air, of the greateft Obliquity of
the Ecliptick, or Latitude of the Place: All
which are requir’d to afcertain the Times of
the Equino¢ctials from Obfervation, and
which being faultily aflum’d, have occafion’d
an Error of near three Hours in the Times
of the Equinoétials deduced from the Tables
of the Noble Tycho Brahe and Kepler, the
Vernal being fo much later, and the Au-
tumnial fo much earlier than by the Calculus
of thofe Famous Authors. es
| Now before we proceed, it will be necef-
fary to premife the folowing Lemmata, fer-
ving to demonftrate this Method, wz.
1. That the Motion of the Sun in the
Ecliptick, about the Time of the Tropicks,
is fo nearly equable, that the difference
from Equality is not fenfible, from five days
before the Tropick, to five days after: And
the difference arifing from the little Inequa-
lity that there is, never amounts to above
3 of a fingle Second in the Declination, and
this by reafon of the nearnefs of the Apo-
gaon of the Sun to the Tropick of Cancer. |
2. That for five Degrees before and after
the Tropicks, the differences whereby the Sun
falls fhort of the Tropicks, are as the verfed
Sines of the Sun’s diftance in Longitude from
_ the Tropicks, which verfed Sines in Arches
under five Degrees, are beyond the utmoft
as a vie nicety
I
204 7." feline Char itd
nicety of Senfe, as the Squares of thofe Arches.
From thefe two follow a third:
3. That for five'days before ant afést the
Tropicks, the Declination of the Sun falls
fhort of the utmoft Tropical Declination, by
Spaces which are in duplicate Proportion,
or as the Squares of the Times by which —
the Sun is wanting of or paft. the) Moment
of the Tropick. |
Hence it is evident that if éhe Shadows
of the Sun, either in the Meridian or any
other Azimuth, be carefully obferv’d about
the Time of the Tr opicks, the Spaces where-
ae
ee
by the Tropical fhade falls fhort of, or'ex= __
ceeds thofe at other Times, are always. pro-
portionable to the Squares of the Intervals
of Time between thade Obfervations and the ~
true Time of the Tropick, and ‘confequently
if the Line, on which the Limits of the fhade
is taken, be made the Axis, and the corre-
fpondent Times from the Tropick expound-
ed by Lines, be erected on their refpedtive
Points in the Axis as Ordinates, the Extre-_
mities of thofe Lines fhall touch’ the Curve
of a Parabola; as may-be feen in the Fi-
gure: Where a, b,c, e, being fuppofed
‘Points obferved, the ‘Lines. aB, bC, cA,
ed, are refpedively: proportional ‘tO. the |
Times of each Obfervation before or after
the Tropical Moment in Cancer.
| This premifed,. we fhall be able to han t
the Problem:of finding the trne Time of the
Tropick by ‘three Obfervations, to this Geo-
“ metrical ones having three: Points in a Paz
rabola A, B,C, or Ay F,C given, together
ane the direftion of the Axis, to find the
si
— Mafcelanea Curtofa. 208
Diftsngeor thofe Points from the Axis. Of
this there are two Cafes, the one when the
- Time of the fecond. Obfervation B is pre-
_ cifely in the middie Time between A and C:
‘In this Café putting t for the whole Time
. between A and C, we fhall have Ac the In-
terval of the remoteft Obfervation A from
the Tropick by the following: Analogy,
/ AS-2ace—bce to 2acei bc::! So ist t
or AE:to Ac the Time of the remoteft
_ Obfervation A from the Tropick.
But the other Cafe when the middle Ob-
fervation is not exadly in the middle, be-
tween the other two Times, as at F, is fome-
‘thing more -operofe, and the whole Time
from A to C: being: put =t, and from A to
& F =, €@-=c,-and b oe be the Theorem
| Bs ttc—bss
will ftand thus - ma Act the Time
2tC—2 Ws Ss.
- fought.
To illuftrate this Method of Calculation it
may perhans be. requifite to give an Exam-
_ ple or two for the fake of thofe Aftrono-
mers that are lefs inftru@ted in the Geome-
‘trical part of their Art.
Anno 1500, Bernard Walther, in the Month
of Fune, at Nuremburg, obferv’d. the Chord
of the diftance. of the Sun from the Zenith
- bya large Parailattick Inftrument of hie
as follows :
Sune 2. “43461: Gunes Se: 44975:
. Fune 9. 44934. and. Fune 12. 44883.
: June 16. 44990 ‘Funei6. 44990.
In
ee Mifitlanes Curiofas
In both which Cafes the middle Time is
exactly in the middle between the Extreams,
and therefore in the former three, a c=§33,
bc=477 and t, the Time between being 14 .
days, by the fir ft Rule, the Time of the.
Tropick will be found by this Proportion, as
_§89 to 8273:: So; t or 7 days to 9 days
2oh. 4’. whence the “Tropick, Anno 1500. -is
concluded to have fallen Fune and. 20h. 2’,
Inthe latter. three, ac is ==) 1e7 7, and
bc=15, and the whole Interval of Time
is 8 days = to t; whence as i99 : to 2067
‘: fo is 4 days to. 4d: 3he 37. which taken
: from the 16th day at Noon, leaves 1144+ 20h.
23. for the Time of the Tropick, agreeing
preliaas the former to the third part of an
our
_ Again, Anno 1636. Gaffendus at Marfeilles,
obferv’d the Summer Solftice by a Gzomon of
55 Foot high, in order to determine the Pro- —
portion of the Gnomon to the Solftitial fhade,
and he hath lett us thefe Obfervations, which
may ferve as an seu for the fecond
Rule. :
'Fune 19. St. N. fhadow 319766 parts, whereof
the Guomon was 89428.
“June 20. 31753
Sune-21. 2 1RkeE
Fume 22.0 208s 31759
Thefe being divided into two Sets of three
Obfervations: each, viz. the roth. 20th. and
22th. and the 1 oth. 21th. and 22th. we fhall
have in the firft three c= 13 and oO 7.
t= 3 days, sco i and i in the fecond c = :
, ; ~~ BD
Mifcellanea Curiwfa. 207
and b= 7, t= 3 and s=—,2.” Whence, ac-
cording to the Rule, the roth day at Noon
the Sun wanted of the Tropick a Time pro-
portionate to one day, as ttc--ssb to
$40-+2b55, that is, aS 110 to'64 in the
firft Set, or 107 to 62.4in the fecond Set ;
that is, 14-17-15’. in the firft, or 14- 174-295’.
in the fecond Set: So that we may conclude the
Moment of the Tropick to have been ‘Sune
rod. 47h. 207. in the Meridian of Marfelles.
Now that thefe two Tropical Times thus
obtain’d, will be found to confirm each o-
thers Exactnefs from their near Agreement,
“appears by the Interval of Time between
them; vz. 14- 2h. 30. lefs than 136 Julian
Years: whereof 14. 1h. Q’. arifes from the de-
fect of the length of the Tropical Year from
the Julian, and the reft from the Progreffion
of the Sun’s Apogaon in that Time; fo that
no two Obfervations made by the fame Ob-
ferver in the fame Place, can better anfwer
each other, and that without any the leaft
Artifice or Force in the management of
them.
What were the Methods ufed by the An-
_cients to conclude the hour of the Tropicks,
Ptolemy has no where delivered ; but it were
to have been wifhed that they had been a-
ware of this, that fo we might have been
more certain of the Moments of the Tro-
“picks we have receiv’d from them, which
would have been of fingular ufe to deter-
mine the Queftion, Whether the Sun’s Apo-
geon be fixt in the Starry Heaven, or if it —
move, What is the true Motion thereof?
It is certain, that if we take the Account
} : of
208 Maifcellanea Curiofa..
of Ptolemy, the Tropick faid to be obferw’d
by Euttemon and Meton, Funii 27. mane, An-
#0 432. ante Chriffum, can no ways be re-
concil’d without fuppofing the Obfervation
made the next day, or Fune 28th in the
Morning. And Prolemy’s own Tropick ob-
ferv’'d in the third Year of Antoninus, Anno
Chrifft 140. was rahe ssi on the 23th and
not the 24th day of Yune, as will appear
to thofe that fhall duly confider and com-
pare them with the length of the Year de-
duced from the diligent and concordant Ob-
fervations of thofe two great Aftronomical
Genii, Hipparchus and Albatam; eftablifh’d
and confirm’d by the Concurrence of all the
Modern Acctracy. For thefe Obfervations
give the length of the Tropical Year, fuch |
as to anticipate the Fulian Account only one
day “in 300 Years; but we are now fecure
that the faid Period of the Sun’s Revolu-
tion docs anticipate very nearly three days
in 400 Years; fo that the Tables of Ptolemy _
founded on that Suppofition, do err about a —
whole day in the Sun’s Place, for’ every 240)
Years. Which principal Error in fo Funda-—
mental a Point, does vitiate the whole Su- |
perftructure of the Almageft, and ferves to
convict its Author of want of Diligence, or
Fidelity, or both.
But to return to our Method, the great’
Advantage we: have. hereby,” is, that any”
very high Building ferves for an Inftrument,
or the Top of any high Tower or Steeple,
or even any high Wall whatfoever, that may.
be fufficient to intercept the Sun, and caft a
true eotea Nor is the Pofi tion of the Plane.
OR
Mifcellanea Carole. : 209
bn which you take the fhade, or that of the
Line therein, on which you meafure the Re-
-cefs of the Sun from the Tropick, very ma-
pterial; but) in’ what wey foever you: difeove er
“it, the faid Recéfs will be always in the fame
- Propor tion, ‘by reafon of the fimalnefs of the
Angle, which is not fix Minutes in the firft
five days: Nor need you enquire the height
-or diftance of your Buildings, provided it be
ivery great, fo as to make the Spaces you |
-meafure large and fair. But it is convenient
*that the Plane on which you take the fhade_
“be not far from Perpendicular to the Sun, at
‘Teaft not very Oblique, and that the Wall
which cafts the fhade, be ftraight and fmooth
‘at Top, and its Direétion nearly Haft and |
Welt, for Reafons that will be well under- .
ftood by a Reader skilful in the DoGrine of
‘the Sphere. And it will be requifite to take
‘the Extream greateft or leaft Deviation of
‘the fhadow of the Wall, becaufe the- fhade
“continues for a good Time at a ftand, with-
“out alteration, which will give the Obferver
eifure to be aflar’d of what he does, and not
be furpriz’d by the quick tranfient Motion
ef the fhade of-a fingle Point at fuch a dl-
tance. The pr incipal Objection i is, that the
Penumbra or Partite fhade of the Sun, is in
‘Its Extreams very difficult to diftinguihh from
‘the true fhade, which will render this Obfer-
vation hard to determine nicely. But if the
‘Sun be tranfmitted through a Telefcope, after
the manner us’d to take his Species in a Solar
‘Eclipfe, and the upper half of the Object- -
‘giafs be cut off by a Paper palted thereon, —
and the exact upper Limb of the Sua be sue
Pp - jul
by ae i, hE tie Re SS) cep
210 Mifcellanea Curtofa.
jut Emerging out of, or rather continging
the Species of the Wall, (the Pofition of the
Telefcope being regulated by a fine Hair ex-
tended in the Focus of the Eye-glafs) | am
affur’d that the Limit of the fhade may be
obtain’d to the utmoft Exaétnefs: And of
this I defign to give a Specimen by an Obfer-
vation to be made in fume next, by the help
of the high Wall of St. Paul’s Church, Lon-
don, of which fome following Tranfaction
may give an Account. In the mean time
what I have premis’d may fuffice to fet others
at work, where fuch or higher Buildings are
to be met with. I fhall only Advertife, that
the Winter-Tropick: by this Method may be
more certainly obtain’d than the Summer’s,
by reafon that the fame Gwomon does aftord
-a much larger Radius for this manner of
-Obiervation-
: Mifeellanea Curipfa. Oe i ae
A Scale of the Degrees of
ss ae Oe
T be Siens and ‘Defcraptions of the
pete Deorees of Heat.
HE Warmth of the Winter
4ir when Water begins to
freeze. This.is known accurately
by placing a Thermometer in Snow
prefs'd clofe together at the Time
_jof a Thaw.
The Warmths of the Winter Air.
The Warmths of the Air in
Spring and Autumn.
The Heat of the Air in Summer.
The Heat of the Air at Nooz in
the Month of July.
The greateft Heat that a Ther-
mometer acquires, by the contact
of a Humane Body ; which is much
the fame with that of a Bird brood-
28 upon its Eggs.
The nearly greateft Heat of a
\Barh, that a Perfon holding his
Hand fteady and immoveable in the ©
fame, can endure for fome time.
17412-/ The greateft Heat of a Bath,
that a Perfon holding his Hand
| ffteady and immoveable in the
me) P a fame,
412 psteetanen Coriohae
ae can endure for fome time. q
| 2° Ori The Heat of a Bath by which J
i melted Wax fwimmipg upon it, 7
| a begins to grow ftiff, and lofe its” q |
a Tranfparency. | 4
L243] Oa ie Ted: Dra Bath by which
| |Wax fwimming-upon it, is. melted §
land preferv’d in a State of Fluidi- ' |
ty, without Ebullution. 4
The middle Degree of Heat, beat .
tween that by which Wax is melt-- :
led, and that which makes Water | 4
, boll, i ,
The Heat . by wien Water is S|
gs to boil vehemently; and a
Mixture of 2 parts of Lead, 3 off)
Tin, and 5 of Bofmuth, cooling, f
begins to harden. f
Water begins to boil witha Heat! 7
of 33 parts, and by boiling, hard=
ly conceives a ‘greater Heat than |
that of 34. parts.
Bat Iron. growing cool, when if
Ihas a Heat of 35 OF "36 parts, ceafes|
to make any Ebullition when warm q
Water falls drop by drop upon it;
as it.does alfo with a Heat of 37
parts, when cold Water falls on it
| in the like manner. . : |
4005 “Io3.) The leaft Heat, by. hich a Mixs |
‘ture of 1 part of Lead, 4 Of Tiny |
land 5 of Bifmuth is liquefied and | |
| _{preferv’d in a State of Fluidity.
48. 3-|. The leaft Heat, bywhich 4Mist |
| ture of equal parts of Tis and Bif4 “|
uth is, aN BRE- This Mixtur@
gr0 io |
ae
a“ |
ee Guta” erg”
growig cool, when it has a Heat
of 47 parts, is coagulated.
ie bs meahe Heat, by Which a Mixture
fot 2, parts of Tin and t of Bifinuth
is liquefied ; as alfo of 3 parts of
Tin and 2 of Lead. But a Mixture
of 5 parts of Tiz and 2 of Béfwuth
(cooling) dees with this Degree of
| {Heat become hard: And the fame
| Icome to pafs in a Mixture of e-
Jes qual parts of Lead and Bifmuth. |
32 “The leaft Heat by w hich a Mix-
Iture of 1 part of Bifmuth and 8 of
ia (Tin is liquefied. Tin by it felt ts
fus’diwith a Heat of 72 parts, and
|. growing cold, hardens with a Heat.
of 70 parts. |
Ae 34+ The Heat by witch Bifmuth i$
ffus’d; as alfo a Mixture of 4 parts
of Lead and 1 of Tin. | But a Mix-
ture of 5 patts of Lead and 1 of
' \Ti when fas'd, and growing cold,
ee hardens with this Degree of Feat.
41 -The ‘leaft Heat by which Lead
is melted. Lead melts with a Heat
of 96 or 97 parts,.and growing
(cold, hardens with a. Heat of 95
eh ers
“114.444 ‘The Heat with’ which Fery Be
hee (growing cool) wholly ceafe
fhinins in-the Night; as alfo, that
-{Fleat: With which (crowing ‘warm ) |
“tthey- firft begin to fhine in the
the Darknefs of the Night, bue
with a faint and Fora Lieht,
fach as can fearce be difcern’d.
Posy’ This
96.
Ay Se
Ree ret AeA HET SHEER Se Te aD LIOA RES a ean
uf [This Heat liguefies a Mixture of 4
136-\43-
192.| 5:
equal parts of Tim and Regulus Mar-
‘is; and a Mixture of 7 parts of
Bijmuth and 4 of the fame Regulus
(growing cool) hardens with the
fame Degree of Heat.
The Heat by which Fiery Bodies
do in the dark Night appear bright
and fhining, but not in the Twi-
light. A Mixture of 2 parts of
Regulus Martis and 1 of Bifmuth,
as alfo of 5 parts of Regulus Afar-
tis and 1 of Tin, growing cool,
will at this Degree of Heat be-
come hard. The Kegulus by it felt,
hardens with a Heat of 146 parts.
~The Heat by which Fiery Bodies,
in the Twilight, a little before the |
Sun’s rifing or after his fetting, do
fhine difcernably.; but not at all -
in the clear Day-light, or at leaft —
very obficurely. han oie
The Heat of a fmall Culinary —
Fire made of Sea-Coal, © burning
freely by it felf without the help
Bellows. The fame is the. Heat
of Iron, as Red-hot as it can be
made in fuch a Fire. The Heat of ©
a fmall Culinary Fire made of |
Wood, is fome little matter grea-
ter, viz. about 200 Or 210 parts. ©
1And the Heat of a large Fire is”
{till greater, efpecially if it be blown -
up by the Bellows. :
Ja
~ Mifcellanea Curiofa. 215
- In the firft Column of this Table are the —
feveral Degrees of Heat, going on in at
-Arithmetical Progreffion, beginning with
that Degree of Heat, which there is in the
Air in Frofty Weather, when Water makes
the firff Advances towards Freezing , beginning
the Account from this, as the loweft Degree
of Heat, or common Terminus of Heat and .
Cold) and fuppofing the external Heat of
a Humane Body to be rated at 12 parts. In
the fecond Column are.the Degrees of Heat
in a Geometrical Proportion, fo that the
fecond Degree is double the firft, the third
double the fecond, and fo on; the firft De-
gree being that external Heat of a Humane.
Body, proportion’d to the Senfe. But now
*tis manifeft from this Table that the Heat.
of Boiling Water is almoft 3 times greater
than that of a Humane Body; and that the
‘Heat of melted Tiz is 6 times, of melted
Lead 8 times, of melted Regulus 12 times,
and of ordinary Culinary Fire 16 or 17 times
greater than the foremention’d Heat of a
Humane Body. : 3
_ This Table was made by the help of a.
Thermometer and Red-hot Iron. By the. ~
Thermometer I found the Meafure of all
the Degrees of Heat as far as that by which.
Tin is melted; and by the hot Iron I found.
the Meafure of the reft. For the. Heat
which hot Iron does communicate to. cold
Bodies contiguous to it in a given time,:
(that is the Heat which the Iron it felf
fofes) is as the whole Heat of the Iron. And
therefore if the Times of Refrigeration are.
ORE ie RP 4 ) taker
Ry SL ee COE a ee
Sit
do en ah AR ae RNa eS SA a Lad
216 Mifeel anea Curt i acme, |
taken aqinils the Deoress of Heat fhall be?
ina Geometrical Propor tion, and ote ena | }
ly may cafily be found by a Table of Loga-—
rithms. © Firft of all: therefore I found by a_
Thermometer made of Linfeed Oil, ‘that if
when the laftrument' was “placed in melting
Snow, the Oil occupied’ a Space of 19000"
parts, the fame Oil rarified by a Heat of the
frft Deptee (that is by that of a’ Humane’
Body) would extend to 16256 parts; and
by the Heat of Water beginning to boil, to.
¥o70§° parts; -and by the Heat of ‘Water
boiling veheinently, £0. 10725 parts’; and: by
the Heat pe melted Tin (cooling, ‘and be-)
ginning to be of the Confiltence of an Amal-
ama) ‘to 11516 parts ; and by the Heat of
the fame Tin when "tis quite barden’d, to
11496 parts.’ Therefore the Oil.was rarified
in the proportion of 40°to°39, by. the Heat.
of a Flumane Body$ and in the: proportion:
of 15 to 14, by the Heat of boiling Water 5°
and in the proportion of 15 to 13 , by the’
Feat of the melted Tin, be: sinning: to come.
to the Confifteace of an Amalgamias and in
the proportion, of 23 to\26; “by. rth “Heat of,
the fame Tin quite hardned." ay ay
AAT Be rechten of Ait with An wn Bet |
gree‘of Heat, was 10 tives greater than that?
of Oil; , and th é Rarefa@ion of Oil nearly” 15,
times greater ‘than that of Spirit’ ‘of ‘Wine ‘i
Now’ th hefe ‘thittes thus found. fa ppofing cn
Degrees’ of Heat'in’ the Oir 4 THe rope
tional’to ‘its’ Rarefattion, and che’ é of 4)
Humane’ Body to’ be 12° ber, froin hence
the Heat Of: Water wheat ; begitis ‘to boil, iy
i A comes
4
a
- Mifcellanea ue oi 2.1 7
“comes to be 33 parts, and when it boils: ve-
nemently, 34 parts; and the Heat of melt-
ed Tin beginning to come to the Confiftence
of an Amalgama, 72 parts; -and the Heat
of the fame, when in cooling "tis: come to —
downright Hardnefs, 70 parts. And having
determin’d thefe Things, in order to find
out the reft, I heated a piece of Iron ’till it
~ was Red- hot enough, and taking it out of
the Fire with a pair of Tongs that were
alfo Red-hot, and I it in a cool place,
where the Wind blew conftantly. Then
putting oe it little pieces of Metals and
various other liquable Bodies, I obferv’d the
times of Refrigeration, "till all thofe melted
€
parts having quite loft their Fluidity, be-
came hal’d and folid- again, and the Heat of
the Iron was equal to that ‘of a Humane Bo-
dy. . Then fuppofing the Exceffes, of the
Heats of the Iron and the liquefied Parti-
cles approaching to Induration, above the
Heat of the Atmofphere founded by the
Thermometer, to be in a Geometrick Pro-
ereffion, when the Times are in an Arith-
metick one; by this means all the Degrees
of Heat were. difcover’d. But ’tis to be ob-
ferv’d that I plac’d the Iron not in a ferene |
and quiet Air, bat in a Wind blowing uni-
formly, fo that the Air which was heated by
the Iron might always be carried away by
the Wind, and a cold Air with an uniform
Motion might Hicceed. in the place of. “it.
For thus, equal parts of the Air were heat-
‘ed in equal times, and acquired a Heat
pene to that of the Iron. But the
\ Degrees
Ee TS ON ORE ROT Ta LEE ue Sob EU EPG ERO ENC Ce sees the eye ay aes
918 Micellanea Curtofa. es
Degrees of Heat found by this Method had
the fame Proportion among themfelves, that
thofe had which were found by the Ther-
mometer; and therefore the Aflumption,
that the Rarefactions of the Oil were pro-
portional to the Degrees of Heat, was a
juft and true one. ; *
™~
i
The 4
id
i}
i
/
s
°
cal cael “hig | i ~ ,
— — Maifcellanea Curiofa. 219
The Properties of the Cate-
| aria.
_ By David Gregory, M.D. Savi-
Profeffor of Aftronomy, and
Fe « Gee aD |
PROP.L PROBLEM.
To find the Relation of the Fluxion of
the Axis, to the Fluxion of the Ordz-
dinate im the Catenaria,
ET FAD be a Catena hanging on the
|_j Extremites F and D, the loweft point
of which (or the Vertex of the Curve) is A,
the Axis AD perpendicular to the Horizon,
and the Ordinate BD parallel to the fame.
We are to find the Relation between Bb or
De’, and df; fuppofing the point 6 infinitely
near to B, and bd parallel to BD, as alfo
DS to BA.
From the Principles of Adechanicks, ’tis
plain that three Powers which are in Equil:-
brio. are in proportion to one another, as
three right Lines parallel to their refpective
Directions (or inclin’d in any given Angle
to them) and terminated at their mutual In-
terfection.
vu And
_. perpendicularly upon Dd, and by which it
— 220 Maifcelanea Curiofa, ees
And confequently if Dd expounds the ab- —
folute Gravity of the Particle Dd (as it will
be in a Catena equally thick) then dS will re-
prefent that part of the. Gravity which ats
comes to pafs that dD. (being by the flexibi-
lity of the Chain moveable about d) endeavours
to bring it felf into.a Vertical Pofition. And —
therefore if Sd (Cor-the Fluxion of the Ordi-
nate BD) be Gonftant, the AGion of the Gra-
vity exerted perpendicularly upon. the cor-
refpondent parts of the Cztexa Dd, will alfo
be conftant, or every where the fame. Let
this Action or Force be expounded by a.
Farther ; From the above cited Propofition
_in Mechanicks, Dds or the Fluxion of the
aE REN GR re Cle peat arg Sg I aa
Axis AB, will expound the Force to be ex-- 7
erted in the direction dD, which is. equiva- —
lent to the former Exdeavour of Dd (by which ©
it tends to bring it felf into a Vertical Pofiti-
on) and 1s fufficient to hinder it.
But this force arifes from: the Linea Gra-
vis DA pulling with the direction. dD,, and.
is confequently (all the reft continuing as be-.
fore) proportional to that Line DA. ‘There-.
fore dd, the Fluxion of the Ordinate, js to
$D,. the Fluxion‘of the Abfeciffe, as the con- »
SSSA areca sah.
Itant right Line 4 to the Curve DA. Q: @
, 7 . 2 a 4 :
, . a EN ‘ iggs ‘ A ore pa Pee! i
4 ; . ee. oP e's ae | ate 4
C; 0. B..Qrobaae ae oe ee
rs ° ; : 2 : ; of states
‘IE the right Line DT touches the Catena- t
ria, and meets the Axis AB produc’d in T, ©
then will DB: BI: (2c 2Deag a.DA,&
Curve. pee
ie PROP.
Bee: Mifcelanea Curiofa. ore
| ee F: I. THEOREM.
: (Fig.34.) If upon the Perpendicular AB.
as an Axis, and the Vertex A, an E-
qailateral Hyperbola AM be defcribd, °
whofe Semiaxis ACS a; as allo upon
the fame Axis and Vertex, a Parabola
“AP whofe. Parameter is quadruple the
Axis of the Hyperbola, and the Ordt-
pate of the Hyperbola HB be always
prodac’d till WE be equal to the Curve
"AP: I fay then, that (making BD ana
BF, equal) the Curve FAD, ia which
the Points F, D, are pofited, is the Ca-
~ tenaria.
Put AB— +; then Bbh=-~, and BH =
N/ sax-l-xx ; ‘whence (from the Method of —
Fluxions) the Fluxion of BH, that is mb =
ax-| sexe | | |
Again, fince the Parabola AP
ec.
‘has for its Parameter 82, BP fhall = i 8ax;
Whence-the Fluxion of BP, that is p=
- BAN
1, Wherefore the Fluxion of the Curve
af DAR | |
| AP
rads Me hi SL Aim
222 Mafcellanea Curiofa. —
AP ( = ae V wpt-| Pn’ ) — f= aa?
| | dana
NV oak? line: ; which is ot to
xt ioe
as appears by multiplying both Numerator
Cena
and Denominator into yh aie Das And fince
LE is every where = == AP thie Fluxion of
9.
Ia :
HF that is me sf, fhall = ————-. But
ee
axfxe
we have hitherto found mb =
| (aos
Therefore sf (the Fluxion of BE the Ordi- .
3 AX
nate Le the Coenanan —
3 and
ty aa 2ax-|-K
confequently the Fluxion of the Curve AF |
ene
fda is, Ff= V sf Let Ve x +3? )
Ax=\- 20d
PS x
is = +——.
Ve le
™ the Plowing Quantity of
which
Mifcellanea Curtofa. 223
° ai | : ; en are
which was fhewn but now to be V azole
And therefore AF = Nate. And ’tis
plain, that the Fluxion of the Ordinate BF,
AX °
or ———-——,, is to x the Fluxion of the
: V 2ax-lox , |
Abfciffe AB, as the conftant Quantity 2, to
the Curve AF; which was the Property of
the Cuatenaria found above. Therefore the
points of the Catenaria are rightly determin’d -
by the foregoing Conftruction. Q: E: D.
won OL. 1
It is manifeft from the Conftru@tion, that
BF the Ordinate in the Catezaria, is equal to
the Parabolick Curve AP, taking away BH,
the correfpondent Ordinate, of the conter-
‘minal Hyperbola AH.
GO ROL. IE.
’*Tis~plain from the Demonftration, that
the Curve of the Curenaria AF, is equal to
BH the correfpondent Ordinate of the con-
terminat Equilateral Hyperbola. For fince
the Fluxions of thefe Lines are equal, and the
Lines themfelves do arife together, it is ma-
nifeft that they are always, and every where
equal. Whence, giving the Catena, AC or a
will be given alfo,.as being equal to the Se-
| maxis
224 Mifcellanea Curwofa,
miaxis of the Equilateral Hyperbola, whofe -
Vertex is A, and whofe Ordinate belonging —
the Abfciffe AB, isequal to the Catena AD.
COROL: 1. a
‘Ail the Catenaria are fimilar to one ano-
ther; as being generated from the fimilar
5.
a
ConftruGion of Similar, and firnilarly pofited
Figures... From whence it follows, that two
right Lines fimilarly inclin’d to the Horizon,
carriéd thro’ the Vertices of the Catenaria, will
eut off fimilar Figures, and proportional to the |
Lines cutting off the Portions of the Cate-
| NATIA. ;
CoRG EL” ae
If the Catena QAD be fulpended on the
points Q and D, which are unequally high,
the part FAD of the Curve remains the fame
as if it.were fufpended by the points F and
D, which are equally high. Fort it is no
matter, whether the point be fix’d to- the
Vertical Plane or not: rat
ratolet eee ioe
| Ny Soe, Fhed
If the force of the Cutesa drawing in the
Dire@tton dD, be divided (as is commonly
known) into. the force as d/ acting with an
Horizontal DireGion, and the force as SD
witha perpendicular Direétion : ‘Then it fol+
lows, that the forceo(in the end of the Ca-
\ ! tena) .
PX
Bia en eg Mee ole Sy
A
: Mifcellanea Curiofa. 225
tena) of approaching dire@ly to the Axis, is —
‘to the force of defcending perpendicularly
in the fame (or that part of the fuftaining
force that acts in the direction BD, is to that
“part that acts in the dire@tion DJ) as the
Semiaxis of the Conterminal Hyperbola AH,
to DA the length of the Catena to the Ver-
‘tex. Whence, the Catena being daveb: this
Ratio is alfo given. And in the fame Ca-
tena, fafpended with different degrees of
Laxity, that Horizontal force, is as the Axis
‘of the Conterminal Hyperbola; fince DA
‘remains the fame, if, the Extremities be
equally high. : |
CORO: VL
_. The Catena placed in an Inverted Pofition
in a Vertical Plane, maintains its Figure and
does not fall down; and fo makes a fine
“Arch or Formx. That is, very fmall hard
flippery Spheres, difpos’d in the Inverted
Catenaria, will form an Arch, no part of —
which will be thruft inwards or outwards by
the reft, but (the loweft Points continuing un-
‘movd) it is preferv’d by vertue of its Fi-
‘gure. For fince the Pofition of the Points
of the Catenaria, atid the Inclination of the
“parts to the Horizon, is the fame, whether
it be in the Pofition FAD, or in an Invert-
ed Pofition, provided the Curve be ina Plane
that is perpendicular to the Horizon, it is
evident that jt preferves its Figure unchang- |
ed, equally in one Pofition as the other.
“And on the other hand, the Cztenaria are
the
Pe Te GS nce eR)
226 Mifcelanea Curiofa. a
~ the only Genuine Arches or Fornixes. And
an Arch of any other Figure, is for this
-reafon only, fuftain’d, becaufe a Catenarta. is”
included in the thicknefs of it. For if it
were very thin, and confifted of parts that
were flippery, it would not be fuftain’d.
From the foregoing fifth Corol. it may be ga-
ther’d with what force an Arch thrufts the
Walls outwards, that it ftands upon; for
this is the very fame with that part of the
force Cfuftaining the Catena) which draws
with the Horizontal direGion. All other
Matters requir’d in the ftrength and firm=
nefs of Walls, that have Arches fet upon
them, are Geometrically determin’d from
this Theory; which are the principal Things
in Building. ee | |
COROL | vit
If inftead of Gravity, any other force were
fuppos'd acting in like manner upon a flexi-
ble Line, the fame Curve would be produced.
Ex. gr. Suppofe a Wind blowing: equably,
and in directions parallel to a given right
Line, the Line thus inflated by the Wind,
would be the fame with the Catenaria. For
-fince all things that were confider’d in Gra-
vity, obtain in this other force, ’tis plain
that the fame Curve will be produced.
PROP
mM ifcelaned Curis fe. a7
PROP. Ill. THEOREM
(Fig. 35.) The Hyperbola AH continuing
- 45 before, if through A be drawn the
- right Line GAL perpendicular to the
Axis AB, and the Curve KR be de-
feria of fuch a Nature, that BK be
_ 4 third proportional to BH and AC,
and to the right Line AC be applied the
Reétangle AV equal to the Interminate
Space ABKRLA; then fhall the Point
F (the Concourfe of the right Lines
HB, VG) be iz the Catenaria. |
2
; : a
For by Conftru@ion BK = —;
2 . V xan x?
wherefore the Fluxion of the Space
ee ih pe : ;
ABKRLA (=BKkb=BKx Bb) = ——----——.
pane
Space ABKRLA
And fince BF = ——-— , and AC
C
is given, the Fluxion of BF fhall =
' the Fluxion of the Space ABKRLA
RATT
AC
Q2, ae
EE SEER ; eer
Se oe
228 Mifcellanea Curiofa.
re But in the Conftruétion et the”
V 2ax-tx 2
foregonig ea tion, a cee of the Or
Axe
ee ax | i Roi 4
dinate BF, was = ———, ‘Therefore.
Wl dash Oe
this Conftrudion amounts to the fame with |
Catenarias PG ame apeee 1 f
‘conoe
As in the foregoing Propofi tion, the Catena: i
ria is defcrib’d from the length of the Para-
bolical Curve given; fo in this, the defcrip-
tion of it depends upon the Quadrature \f
the PURSE in vee x7y” <= a* ome 2axy”. i |
: : ; ; ~ os olen i
sah . ee BaD EVE : a ¥ :
Pi 4 4 a .
* WV =
2ax-|- a?
- ot lomemeee A q : :
‘ Ne eet OALLS
, ames {
& aaa
j roe
& a me he le a
& yr
in £ & ee aaNet enti, erie ellen Ne mlirete ds ale gD |
ate Fs
Pen.
ae ee ‘
) om ~ : i
artes Ge on } ~ > Ke
ne & = wh y “ ig wi had ak. 03
Cas . - “e Q
oan ~ - > iy
a | : e
Ps eS a nd < PREZ rik’
Mpelbhics OPA A NN By ETS teens Oh artes tt yn a ON
ro
tee
oa - ry
. a -“ rd
oat
~MifceHanea Curiofa. 229
PROP. Iv. THEOREM.
Cg. 36.) The Space AGF contaiw’d an-
der the Catenaria AF, and the right
_ Lines FG, AG, parallel to AB, BF,
boas equal to the Rectangle under the Se.
- wmiaxis AC, and DH the difference of
the Ordinates in the wich emai ana Ca-
tenaria.
ro ‘DH C= BH — BD = = by Prapofi
ax-eeie an
Coe Ths a aa se)
| ae — V aaa-| x? |
ex : : age : a
ee. Wherefore the Eon of the
ae
Redangle under re given Line AC and DH
wae ax
bias RG)
NE suse ea A Sige | ae
= the Fluxion of the Space AFG. And
fince thefe Figures do arife both together, it
follows that the Rectangle under AC and
pa. is equal to the pase 2 AGF. Q: E: dD.
iQ + COROL
—
230 «©Mifcellanea Curiofa.
EQROEL
Hence it follows that the Space FAT
comprehended under the Catenarsa and Hori-
zontal Line FD, is equal to the Rectangle
under FD and BA, lefs the Rectangle un-
der cither Axis of the Hyperbola AH, and
DH the excefs of the right Line BH or the”
Curve AD, above the Ordinate BD.
PROP. V. THEOREM.
(Fig. 36.) If to the right Line AL Ge ap-
plied the Reétangle LE, equal to the
Hyperbolical Space ALH, then E will
be the Center of Gravity of the Cate-
navia AFD, | ; /
Let the Curve FA be conceiv’d to be li-
brated upon the Axis GL. Then (from the
Doétrine of Centers of Gravity) it is mani-
feft that the Afomentum of the ponderating
Cvrve FA is expounded by the Superficies
of an upright Cylinder erected upon FA,
and cut off by a Plane, pafling through GL,
and making an Angle of 45° with the Plane
of the Curve. And the Fluxion of this Su-
perficies or FA x FG, is equal to the Floxion
. of the Space ALH or BH x HL; _ becaufe
FA, BH, as alfo FG and HL, are equal.
eae | une And >
Nope Sh Tein 74
Beene ae
Bee Vl
ri
, Mifcelanea Curiofa. 131
And confequently (fince they arife together) —
the faid Superficies of the upright Cylinder
is equal to the Hyperbolical Space ALH.
Which therefore divided by the Pondus it
f€lf AF, or its equal the right Line AL, gives
the right Line AE, for the diftance of the
Center of Gravity from the Axis of Libra-
tion GL. So that the point E is the Cen-
ter of Gravity of the Curve FAD, lying e-
- qually on both fides the Axis. Q: E: D.
COR OL:
.. The Spaces ABHL, BAH, and AFG, are
In Arithmetick Proportion. For the Fluxion
Be i : Vi alee ol
of the Space ALH is (= ——— ¥ X=
V rax-|x?
ax-l-x*%% = 2ax-\-a2? —ax Xx x
— a ae oe
MV roared? 2a x?
——=
ed
i -— AX ie
oF eee) = to the Fluxion
, Vaax-bx? |
of the Space BAH lefs the Fluxion of the
Space AGF, by Propofition IV. And fince
thefe three Figures do arife together, it fol-
dows that BAH — AGF = (ALH=) BL— ~
BAH. Wherefore 2BAH = BL -| AGP.
B 3H Q4 Whence
+ Rae TS
a
232 Mife ellanea Cutiife.
Whence ’tis plain that the Spaces BL, BAH, q
and AGF, are in Arithmetical Proportion.
COROL. Tl.
| The Center: of Gite of the Bhinatia! 4
is the Loweft of all thofe Lines that have ‘
the fame Termini, and are of the fame |
length. For a heavy Body will defcend as
far as it can, And fince the Figure it felf
defcends as much as its Center of Gravity —
defcends, ’tis manifeft that a flexible heavy |
Line, will difpofe, it felf in fuch a manner,
as that its Center of Gravity may be on :
-er, than if it affum’d any other” ee
And from this one Property of fach a Line, | ’
all the reft may ealily be hcber an; 4
Ee OROL. Hf.
If there be upright Cylinders ere@ed upon — '
any fort of Curves, that are of the famed
length, and have the fame Termini D and F,
with the Catexaria FAD; and thefe Cylind
ers be cut by a Plane paffing through DF;
then the greateft of all thefe Superficies thal
be that which ftands upon the Catenaria.”
For thefe Superficies (if the Angle contain’d ©
under the Planes he half a right one) divi-
ded by the Curves (which in the prefent
Gafe are all of the fame length) give the di-
ftances of the Centers of Gravity from the
right Line DP. And fince this diftance is
greateft in the Catenaria <becaufe of the
aay Defcent of the ‘Center: of Gravity) —
| : therefore -
— Mifcellanea Curiofa. 233
therefore the Cylindrick ‘Superficies. fhall
there alfo be greateft. Laftly, Becaufe the
fame is to be faid of Cylindrick Superficies
cut off by a Plane that make any Angle with
’ the Plane of the Bafis, as is wheu the faid.
Angle is half a right one; the Truth of |
what was alflerted is evident univerfally.
i
LEMMA.
Cig. 37.) Any Curve as AFQ, defirib'a
. by the Evolution of another Curve KU,
af apon any Ordinate, as FB (at right
Angles to the Axis AB) be let fall per-
pendicularly UR, from the corre{pondent
Point U in the Curve KU; then (the
Flaxion of the Axis AB continuing
the fame) fball the Fiuxion of the
Fluxion of the Ordinate BYE, the —
Fluxion of the Curve AF, and the right
Line FR be continual Pr oportionals,
Let the Lineola Ff be produc’d ’till it meets
the next Ordinate Wein oe And becanufe by
the Hypothefis Fs = fw, alfo fhall of = Ff,
and confequently oo fhall be the Fluxion of
fs, that is the Flxion of the Fluxion of the
Ordinate. Farther, the Triangles oo f, fFR,
are Equiangular, becaufe oof = its "Alter-
Maret ER, and. fog (Kir csi ER, bes
caufe their difference R fr is as nothing in
refpe&t of either of them, fince Rr is no-
thing in refpeét of fr. And therefore oo:
soa fF: FR; butef=fF, fince they oe
er
cs wie i mS vy ‘TPR ea,
934 © ©Mifcellanea Curiofa.
fer but by the Fluxion of either. There-
fore alfoog:fF:: fF: FR. Q: E:D.
- PROP. VI PROBLEM.
(Fig. 37.) To find the Curve KV by the -
Evolution of which the Catenaria ts de
feribd. - eka
x : i oe
ES NESE SBT OE
WS
sae ie
Let (as before) AB=x, BF=y. Then by
te se ax Rane
Propofition Il. j= eg Or 2axy* -|-
Voaax-Lsce
xxy? ar atx. Wherefore (by the Newtonian
Method which now genérally obtains) aaxy?
as ryy 2 ex y? -|-26 y (= oarws which
~ becaufe of x = 0, fince the tconftant x has
no Fluxion, is) =¢. Therefore y = :
Seca axy — LY a-|-2x pg
ae
map
2ax |e 2ax-|-% Ae open
| | | V rax-bex
CR Te
ee go CR
| Mh aan | gags
(For the Sign — before the Quantity Y% de-
| ~ notes.
putting inftead of y its Value
j
es eRe eee
Ser sidnly. 9 9." a es
Rens BPA as
— Maifcellanea Curtofa. 225
-- notes only the place of the point R, with —
refpect to F, to be oppofite to the place of
the point F, with refpect to B, when the
Curve AFQ is concave towards: the Axis
testi
AB.) And Ff (by Prop. i) =--—- :
es | has oe
Wherefore (by the foregoing Lemma) FR =
(== abe xx? 2ax[-xx x V aax-L-sw
ONY 2ax--xx a-l-x x ax?
ap CERESET
« V2an-ae, Again, becaufe of
ae) a-|-x
a
the Rectangular Triangles Fs f, FRU, ha-
ving the Angles f Fs, UFR, equal to one
another (becaufe UFs is the Complement of
either to a right Angle) we have Fs: sf::
a
axe ah xV aan--xx
FR: UR,or+: mes ee
a
/ 2ax{-xx
UR, which therefore is = 4-|-x. Therefore
_ the Nature of the Curve KU is fuch, that if
panes 0 Pe
a-|-x x V/ 2ax\-2xe
: AB = *, FR fhall = er and
| 3 a
UR =a}x. Q: E: I
COROL.
Ra M. efeelanen Curiofa.
| COROL. E-
“AC: CB:: BH: BH: FR. poe this is
the Property of the right Line FR, fe was:
found juft tow.
COROL. Tt.
The right Line CB is = the nen Line
Bl or UR. For each is = a-|-X-
“CORO LTHEG jaa
The Evoluent Line UF is a third Propor-
tional to AC and CB. For becaufe of the
fimilar eid fis, UFR, it iss F: F f
palin fos
Pua 3 ©
: FR: UF ; ore:
(et Preenag J
Vi2as-t-xe |
—=—eSe
a-\-% x ‘/ 3 AX-\-2N
et
at
y
[x |
<= ———-, Whence a: a-\-x: f atx: UF 9
phick | is the Radius of the Circle that has
the fame Curvature with the | Capenanis at the
eee F.
COADL.
——: UF, which is therefore
¥
ee
tea
$35
ca
| Mifeellanea Curiofa. 237
COROL. IV.
‘When the point F is in A, or when the
Vertex is defcrib’d by the: Evolution, that i is,
_ when x =o, then the Value of the Evoluent
_Line (or the Radius of the Curvature) UF
(which in this Cafe coincides with KA) viz.
Zz
ae
» becomes only 4.~ That is, the point
a
K where the Curve UK meets the Axis, «1S
as ‘much above the Vertex of the Catena A,
as C is below it. Whence the Diameter. of
the Catena at the Vertex, — ‘is equal to the
Axis. of the Conterminal Hyperbola. AH.
And confequently the Cateza AD and the
Hyperbola AH, have the fame Curvature in
the Vertex A. For it 1s known that the
foremention’d Circle has the’ fame Curva-
ture with the Equilater: al Hyperbola AH, in
the Vertex’ A:° But ‘this follows alfo. from
the Property ofthe Catenaria, demonftrated
at Rropefitian Ihe For the Nafeent PH Or CAP
= the Nafcent BP —) \ eee is double the
Nafcent BH or (V 2ax-L ae, that is, xv va-
nifhing, when x is very fmall) /24x. And
therefore the fame point is as well in the
Nafcent Fyperbola, as in the’ Nafcent Czre-
naria,.that is, the.one is coincident with the
other at their firft arifing, and confequently
thefe Curves have the fame Curvature at the
erat ECe ae alten
ss ae VS RHEE R OT.
Saar A BN oat ye
gaunt iy PART PSE Rees A gy ae Ten tt IL BHC RM CRON IL to” SOP a me ea (ve tALY a) ~ SS STIPE hth Sieg ERP Ps AP ae a Te
See Mifvellanea Curiofa. Ee
0 ctl pina EOE
COROL. V. |
The Curve KU is a third Proportional to
the right Line AC, and the Curve AF or the ,. —
right Line AL. For from the Property of
the Evolution, KU = (UKA—KA=VF—
ax a? —|-2ax—x?
a a
ae 2. And therefore, 4: V xax-fe i :
V xax-Lex s. KU. But (by Cor. IL Prop. II.)
V 2axlacx — AF. ‘Wherefore AC: AF-::
AF: KV.
eras COROL VL
‘The right Line KI is double of AB. For
fince Bl = (BC =) CA-+AB, alfo Al fhall
— CA-}2AB. But AK=AC)(by Gor. IV. of
this Propofition.) Therefore KI=2AB.
3 COROL' Vil. )
The Reétangle ACxBR is = to double the
Hyperbolical Space BAH. For FRxAC =
(apex V2axfwx “Zz a-t-x« Nf 4 ake) tee inca
: a : Tits »
+
xXx een xx ax A) Sicse ies = ABx BH
: . --AC
Maifcellanea Curtofa. 229
-|-ACx BH=) ABx BH-|-ACx BD-+-ACx DH.
Wherefore FRx AC—BDx AC (thatis, BRx
AC) =ABxBH--ACxDH. But (by Propo-
fition1V.) ACXKDH=Space AGF. Therefore
BRx AC= (ABHL+-AGF= by Cor. |. Propo-
_ fition V.) 2BAH. |
PROP. VI. THEOREM.
(Fig. 37.) If iz the Logarithmical Curve
LAG (whofe Subtangent HS, given, w
equal to the Line a, determin’d as at
Cor. II. Prop. II.) be taken the point
A, whofe diftance AC from the Alymp-
tote HP, is equal to the Subtangent
HS; and from the points H, and P
(taken at Liberty in the Affymptote,
and equally diffant from the point C)
be erected the Lines HL, PG, Ordi-
nates to the Logarithmical Curve, the
half Sum of which is equal to HD or
PF: Then the points D and ¥, {hall be
pofited in the Curve of the Catenaria,
_correfponding to the right Line AC.
Let AB be put = x, and confequently CB
or DH the half Sum of the Ordinates HL,
PG, will = --x. Let the half difference of
them be put = 7; whence HL = 4-|-x-|-y,,
and PG = a-+x«+—y. And fince from the Na-
- ture of the Logarithmical Curve, CA isa
mean Proportional between them, aa fhall =
AA
re
240 © Mifcellanea Curiofa,
at NAO US
aa ve 2ax |X — Vy whence y V2ax-bow.
Confequently HEL = a--x-|- V2ax-|-ax, and
PG —aA —|- x ede eeeed ba Wherefore the
Fluxion of HL, or lm, is
. le eae 2ax-|-wx
: Woe as :
fimilar Triangles JmL, LHS, ’tis LH: HS::—
lm: mL; whence mL or dd the Fluxion of
And becaufe of the
aX
Lis EAE is, the Curve
: NV) rax-[ox |
AD, generated after the foregoing manner,
from the Logarithmical Curve, is of fuch a~
Nature, that if the Axis be x, and its
BOS is
Fluxion xe, the Fluxion of the Ordinate : BD
Ax
ig’ hte See es ie ee Property of
ere ae oa
the Catenaria correfponding to the right Line
a, as was demonftrated at Prop. 1.’ There-
fore the Curve FAD defcrib’d as aboye, is
this very Catenaria it felf. Q:E: Do
a. 4
‘ Maifcellanea Curiofa. me
Pare OL LARTIES.
COROL. Lo:
As the Catenaria is defcrib’d by the help
of the Logarithms, fo on the other hand,
by the help. of the Catenaria (a Curve pro- ~
duced by Nature it felf) the Logarithm of
any given Number, or rather of any given
Ratio, may be found. As if, putting CA=1,
whofe Log.=o; the Log. of the Number
CQ, or of the Ratic between CA and CQ,
were fought. Let CV bea third Proportional
to CQ and CA, and CB the half Sum of CQ,
and CV; then an Ordinate to the Catenaria
from the point B, viz. BD, will be the Log.
fought.
COR OL. fl.
Vice verfa, if giving the Log. CH or CP,
the carrefpondent Number HL or PG, or
the Ratio of HL to CA, or PG to GA, be
fought. From H or D ere¢t a Perpendicu-
lar meeting the Cateza in D or F, and in the
Horizontal Line AR, take CR==HD or PF,
or CB. And then will AR be the half dif-
ference of the fought Lines LH, GP, as HD
‘the half Sum of the Extreams leffen’d by
: R
or CR, is (from the above demonftrated Pro-
perty of the Catenaria) their half Sum. For
in three Quantities Geometrically Propor-
tional, fuch as are HL, CA, PG, the Square of
the
242 Maifcellanea Curiofa. 7
the Square of the middle Term, is equal to ©
the Square of the half difference of the Ex- —
treams. And confequently CR-|-AR, and
CR—AR, are the Numbers HL or GP,
agreeing to the given Log. CH or CP.
COROL.: Fi.
It is plain from the Demonftration, that
as HD the half Sum of the Logarithmical
Ordinates HL, PG, being applied at. right
_ Angles to CH, is an Ordinate to the Care-
naria, fo alfo the half difference of the fame
HL, PG, applied at right Angles to CA
in B, is an Ordinate to the Equilateral Hy-
perbola, whofe Center is C, and its Vertex
A; and confequently (by Cor. I. Prop. IL).
<== the Catena AD. For y = MV aaie ves
and fince it was fhewn in the foregoing Co-
vol. that AR is alfo the half difference of —
HL and PG; ’tis plain that AR is = the
Portion of the Catenaria AD. From whence
by the way, we may obferve a Method, how,
from the Catena AD given, to find C the
Center of the Conterminal Hyperbola, or
the point in the Afymptote of the Logarith-
mical Curve GL. For taking AR= the Ca-—
_ tena AD, and joining the points B, R, from
the middle of BR ere& a Line perpendicu- .
lar to it, which will meet BA the Axis of
the Catena produced, in the point C, fought.
G | Which Is evident, fince thas CR will = CB.
COROL.
Mifcellanea Curiofa. 243 |
BOR OLS IV.
Hence alfo it follows that if the Angle
BDT be equal to ACR, the right Line DT
touches the Catenaria in D. For then it will
be (in the fimilar Triangles DBT, CAR) |
ie. bes. CA: AR, of CA: Cutve AD
which is = AR. And confequently DT
touches the Cutenaria, by Corol. Prop. I.
QOROL: ¥V.
It follows alfo that the Space ACHD =
the Rectangle CAxAR. For becaufe (by
Prop. IV.) AYD=CAx (AD—BD, —=AR—
AY, by Cor. III. of this Prop. —) YR; the
thing is manifeft. And fince CA is given,
*tis plain that the Space ACHD is as the
Curve AD, and the Fluxion of the former
Hd, as the Fluxion of the Latter Dd.
COROL. Vr
__ If through the point K where CR cuts HD,
we draw KZ parallel to PH, meeting AC in
Z, and tdke CE = eee 3 then will E be
the Center of Gravity of the Curve FAD.
Imagine an upright Cylindrick Superficies
erected upon FAD, and to be cut by a
Plane pafling through PH, and making an
Angle of 45 with the Plane of the Curve
FAD. This Superficies, will expound the
x hoa R 2 Momen=
2 oy ee ee ae
ee a 3
244 — Mifcellanea Curiofa.
pa ee of the Curve FAD librated on
the Axis PH; an@¥its Fluxion is DHxDd-\- —
axe
PF x Fi=2BCx AD=3a p28 \-24 x — =
| Miata <
2a? pa Saw |- 22a wae
Gerpcemm nese OS eee Teese Gees Ey
V 2AK-|-X Vaart [-w3e
a* x-|-axee Zaxxl-2e? x
-|-——-_---—-; the Fluent of
ee a Vague!
Which, ax BD-- av 2ax-|-xx-|- aN sax = esi
‘CAxBD-| CBxAD. Wherefore CAxBD-|-
CBx AD= (Cbecaufe it arifes together with
it) to the foremention’d Cylindrick Superfi-
cies— the Momentum of the Curve FAD
with refpect to the Axis of Libration PH.
Whence the diftance of the Center of Gra-
vity of the Curve FAD from the ee C,
is CAxBD-} CBx AD 4 1 CAxBD -
QADa rut oa y ADanom
Farther, becaufe of ZK. parallel to AR, tis
AD: BD:: (AR: ZK: :) CA: CZ, whence
GZ — CAxBD 4 and therefore CE which by
Conftructiion is = 3 BC-|- 3 CZ, ‘fall oa
aige +-2BC. That is, the Center of
| Craig of the Curve FAD, and the point E
: determin’d
| Mifcelanea Curtofa. 245
determin’d by this Conftru@tion, are equally
diftant. from the point C. But they are alfo
pofited in the fame right®Line, and towards
the fame parts, and therefore they coiacide
with one another. This G@oincidence of the
point E as determin’d above, with the Cen-
ter of Gravity as found at Prop. V. may be
thus /ysthetically fhewn. By Cor. I. Prop. V.
2BAX = AYD-|- BAx AR. Whence AH -+
2BAX = (ACHD -|- BAx AR = by Cor. fore-
going) ARxCA-\|-BAxAR;3 that is, BDx
AC-|- 2BAX =ARxCB; or BDx AC = AR
xCB—2BAX. Whence BDxAC-|- ADx
BC= (ADx BC -\- ARx CB — 2BAX = 2AD
xBC—-2BAX =) 2ADx AC -|- 2ADx AB —
2BAX. And-dividing by 2AD, we have }
VEDY AR inn rac |: ABKXAD—BAX
AD |-z BC = (AC-| aan =
CA -|- pee But ee is the diftance of
| ARS AR
the Center of Gravity of the Catexa from the
Vertex A, determin’d at Prop. V. and confe-
quently, according to the 5th Propofitiow CA
wats ete is the diftance of the point EB from
BD 7
C; now 2 shies -|- 3 BC, is the diftance
of the point E alfo from the fame point C ac-
cording to thisCor. Whence’tis manifeft that
thefe two Determinations of the point EB
; X
amount to the fame; becaufe CA -|- a =
1 BDx AC 4456.
aes ie | a
pore aial > ae ae COROL.
hd
246 “Mifcellanea Curiofa. ‘
CORMOL. VIL
~The Center of Gravity of the Space
PFADH, is in I the middle point of the
right Line CE. For fince the Center of
Gravity of the Fluxion of AD, or Dd, and
Ff, is twice as far diftant from PH, as 1s the
Center of Gravity of the Fluxion of ACHD,
or DHhd, and FPpf; and Dd-+-Ffx AC 1s =
DdhH-\-FfPp; ’tis plain that B, the Center
of Gravity of the Fluent FAD, is twice as
far diftant from PH, as I, the Center of
Gravity of the Fluent PFADH. But this
~ may be yet fhewn otherwife according to the
Method us’d before. Imagine an upright
‘Cylinder to be -ere&ted upon the Figure
PFADH, and to be cut off by a Plane paf-
fing through PH, and making an Angle of
4§ with the Plane of the Bafis. This Solid
will expound the AZomextum of the Figure -
PFADH librated on the Axis PH. And
the Fluxion of this Solid or A¢omentum (viz.
the Solids erected on the Bafis PFfp, and
HDdh) is produced, by multiplying the A@-
mentum of the Fluxion, or the Fluxion of
the Atomentum, into ; AC given. For by
Cor. V. of this Propofition HDdh=Ddx AC,
Wherefore the Fluent Adomentum it felf, is
produced by multiplying the Alomentum of
the Curve FAD with refpe& to the Axis
PH (as determin’d at Cor. foregoing) vz.
CAxBD-|-CBx AD into ; AC; which will
therefore be ACx ACxBD-|-4ACxCBx AD,
Aud confequently if this be divided by the
a pes le Oe ee librated
ie The *, .
4
\
~~
CO Mifcellanea Curtofa. 247
librated Figure PFADH (=2CAxAD, by
Cor. V. of this Propofition) there will arife
(for the diftance of the Center of Gravity of
the Figure PFADH from the Axis PH) 4
CABID HL 2
Za -|-4 CB 5 which. 1s — 3 CE. deter-
min’d above. ;
Pee OR OL, VII.
If through the point N where DT the
Tangent to the Catezaria in D, cuts the
Line AR, be drawn a Parallel to BC, meet-
ing in O a parallel to AR drawn through E ;
then will O be the Center of Gravity of
the Curve AD. For by Cor. 6. the Center
of Gravity of the Curve AD is in the right
Line EO. But it fhall be demonftrated to
be in the right Line NO ; and confequently
‘that O it felf fhall be the point. Let DA
be conceiv’d to be librated upon the Axis
HL; then the Afomentum of this is the
Curve DA multiplied into the diftance of
“the Center of Gravity from HL. And
wity) = Waar lex K ee
confequently its Fluxion = DA x Hh (Hh
being the Fluxion of the diitance of the
Axis of Libration from the Center of Gra-
ax
—=ax. -And
| MV 2ax-|-x?
| therefore the omentum of the Curve DA,
with refpec to the Axis HL, is—= 4x. And
-confequently the diftance of the Center of
Gravity from the fame Axis, is 2x divided
R 4 by
&
“~
248 Mifcellanea Curiofa.
by AD, or SAO But becaufe DT -
touches the Catenaria, by Cor. 4. of this Pro-
pofitien, the Angle BDT, or DNY=ACR, |
and the Angles at A and Y are right ones,
therefore in the Equiangular Triangles”
RAC,” DYN,>. ’tis’ RAY AGr. Bae 7S
whence YN = eee that is Yas, the —
- diftance of the Center of Gravity of the
Catena AD from the Axis HL; or that Cen-
ter is in the right Line NO. i
COROL IX.
If through the point I be detera” a right
Line parallel to AR, meeting ON produc’d ©
in W; then W fhall be the Genter of Gra-
vity of the Space ACHD. For by Cor. 7
the Center of Gravity of the Space ACHD,
isin the right Line TW, but it fhall be de-
monftrated alfo that °tis in NW, and confe-
quently W is the point. For (after the
fame manner as in Cor. foregoing) the Fluxion
of the Afomentum of the Space ACHD ponde-
rating upon the Axis HL, will be fhewn to be
(ACHDx Hh=ACx ADx Hh= =) Aa
aX
= ax. And confequently the —
Vax} foie :
Momentum of the Space ACHD, with re-
{pect to the Axis HL, is the Fluent of this _
“Fisxion,
— Mifcellanea Curiofa. 249:
Fluxion, az, that is, 27x. This therefore
divided by the Space ACHD, or
ee gives the diftance of the
_ Center of Gravity (of the Space ACHD)
| ; Be
from the Axis HL, which is = —
le luxe
- And therefore the Center of
- BGs DY
PS CAR
Gravity of the Space ACHD, is in the Line
NW. And fiom thefe two laft Corollaries,
is found the Center of Gravity of any Por-
tion of the Catena, though not reaching
the Vertex A, or alfo of any Space com-
prehended under any Portion of the Cate-
maria, and any other right Lines befides
thofe aforefaid.
z=
COROL.. X.
Hence are meafur’d the Surfaces and So-
lids generated by the Rotation of the Ca-
tena (or a Space comprehended under it,
and a right Line) about any given Axis.
For a Figure generated by fuch a Rotation,
is (as is vulgarly known) equal to the gene-
nerating Figure multiplied into the Periphe-
ry defcrib’d by the Center of Gravity in the.
Rotation, which Periphery is given, fince
the Radius or Diftance of the Center of
Gravity from the given Axis, is given. Thus
if the Catena AD roul’d about the Axis oo
as ined Aa
aso © Mz fcellanea Curtofa.
then a AN is the Periphery deferib’d by the
Center of Gravity O oF denoting the Ratio
of the Periphery of a Circle to the Radius) oy
and confequently the Surface generated by
the Rotation of the Catexa AD = Fx
ANx AD=) ~ x ANx AR. That is a Cir-
cle, the Square of whofe Radius is double
the Rectangle RAN, will = the Surface ge-
nerated by the Rotation of the Catena AD
about the Axis AB. After the fame man-
ner the Solid generated by the Rotation of
the Space ACHD about AC, may-be fhewn ~
- to be equal to a Cylinder, whofe Bafis is the
foremention’d Circle, and its Altitude=AC.
_ Thus alfo the Surfaces and Solids produced
by the Rotation of thefe Figures about any
other given Axis, are meafur’d. For givin
the Center of Gravity, they are eafily dif-
coverd. | :
Of
—Maifcellanea Curiofa. - 251
Be Of the Quadratures of Geometri-
cally wratinal Figures.
By J. Craig.
- ET ACF (fig. 38.) be a Semicircle,
whofe Diameter is AF, ADE a Geo-
metrically irrational Curve, whofe Ordinate ©
BD cuts the Semicircle in C. The Quanti-
ties may be noted thus; The Diameter AF
=a, 100 Abicifle AR = y, the Arc AC =
v, the Ordinate BD =<: And let 2 =rvy”
a General Equation exprefling the Nature of
the Geometrically irrational Curves ADE, in
which rv denotes any given and determin’d
Quantity, and # an indefinite Exponent of
the indetermin’d Quantity y. I fay the Area,
: roy cata
ABD = — qv-[ V aay — yy x
ra 2nra® —\-ra®
ee eee OP
waht nx n-|-1|
AAX2I—t BX 2M —3
: Ca a SOIREE rae si
i—1 Pe
ACK20—J AD 27i-—7
— i—
y" 4 -|- cemnenraccemanaman) J 5
in—3 woth
akE%x 2m
252 Maifcellanea Curiofa.
AEX 2N—9 | |
ane
In this Infinite Series, thefe things are to
be taken notice of :*(1-) That the Capital
Eetters A, B, C, D, EB, @e dencte theta.
efficients of the Terms immediately pre-
2nraa -\-raa
B=
‘ceeding them, wz. A= se
ey eae aX n-\-1 Xn-|-1
aAkin—1 aB%2n—3 - . at,
, and foon. (2.)
ete, OG
a— 1 ND,
That if the Exponent 7 be an Integer and
-Pofitive, or equal to nothing, or if 2” be an
odd Number, then the Quadrature of the
Space ABD may be exhibited by a finite
Quantity: The Seriesinthefe Cafes breaking
off. (3.) That q denotes the Term laft break- —
ing off (4.) That all thofe Figures in which °
the Series is broke off have one Geometri-
cally Quadrable Portion very eafily affign-
able from the Series it felf, vzz._if you make
E I
the Abtciffe yr *F!-[-ng} gl; there
will arife a Geometrically Quadrable Areaan-
fwering to this Abfcifle. (5.) That only the
Irrational Terms V xay—yy is to be multi-
tiplied into the Terms following it.
E xample
“Mien Cw | 252
Example ¥ I.
Let z= v, becaufe in this Cafe r= 1,
| ra
n=, therefore — is the Term laft
n-\-1|?
breaking off wherefore q = 4, whence ABD
= vy —av-|-a V 2ay—y2: And confequent-
ly if (by Note 4.) you take the Abfciffe
y = 4, that is, if the Ordinate pafs through
the Center of the Circle, there will arifea
Geometrically Quadrable. Portion fitting it,
viz. Area = a*, that is, the Square of the
Radius. :
i dann TI.
¥ vy | ak
Let z==—. Becaufe in this Cafe r —
-2na* as ra?
== 1, therefore —-—
ny n—|~-1) a?
d
y"—* isthe Term
| oe Le
a breaking ot jes qg == —; whence
rs os we E A 4 Z
, es 3av yr a.
Ae ay oo oT te
Vay —y 2 and
confequently, if uae Rote pe ‘you take
y= RAE 39% there will arife a Geometrically
) Quadrable
284 Mife ellanea Curiofie.
Quadrable Area fitting this Abfcifle, .viz.
area = Vi Hide -|- oo
Heal HHI.
Let = a
In this Cafe r= ; —-, Bk,
aa
; a 2i——T yoo '
therefore —————— _ y"? is the Term laft
Yom]
breaking off, therefore 7 = x : whence by
Infinite Series, will ABD =
6uy3—1 5a? v-|-2ay?-|- ‘3a y- _L1s oan
EE NS NES Se
ee 3
18a?
And confequently, if (by Note 4.) you take
y= a/ i > there will arife a Geometrically
Quadrable Area fitting this Abfcifle, viz.
aay" -|- 5a 'y rh 15a? |
te
x aay sae y? 5
Area =
184
Secondly, Let ACF (Fig. 39.) be a Pow
bola, AE its Axis, A the Vertex, and (Ba)
the Latus Reftum. And let. ADG bea Geo-
metrically irrational Curve, whofe Ordinate
BD cuts the ParabolainG. Let thé Abfcifle
AB= y,
— Maifcellanea Curiofa. 255°
AB =y, the Ordinate BD = z, the Arc of
the Parabola AC =v. And let the General
Equation exprefling the Nature of Infinite
irrational Curves be this, Z = rvy”, in which -
_ry denotes a given and determinate Quantity,
and z an indefinite Exponent of the indeter-
mind Quantity y. I fay the Area
alt prea
wet) ese qu -|- V aay ye
i n-\=1 2
ag n-\-t ad a ne
n-|-2X%2-|-4 p-oxnl1 2)
Reg pt SARIN! yaa
nx n-|-2Xn-|-11 3 See !
' ABR Ii—3 yy 9 AC KIN—F§
eee ag
In this Series ’tis to be noted: (1.) That.
the Capital Letters A, B, C, cc. denote the
- Coefficients of the Term preceding them.
(2.) That if the Exponent z be an Integer
aad Pofitive, or equal to nothing, or if 27 be
an odd Number, then the Quadrature may
be exhibited by a finite Number of Terms;
the Series in thefe Cafes breaking off. (3.)
That--|- ¢ is equal to the Term laft breaking
. off (4.) That of the Terms multiplying the
Quantity VV 2ay -\-y?, the laft breaking off is
to be doubl’d. (5.) That all thofe Figures
in which z is an Integer, Pofitive and an odd
Number, or more generally, all thofe Fi-
gures in which the laft Term kaa hind
: as
956 © Mifcellanea Curiofa, —
has an Affirmative Sign or+|-, have one —
Geometrically Quadrable Portion, and aflign-
able from the Series it felf, by taking the
Ab{fcifle as in the fourth Note of the pre-
ceding Series. | mete? s
Example d.°
‘Let z — v, becaufe in ‘this Cafe r 4; -
n—o, therefore the Term laft breaking off
is — a es whence -|-4 = (by
n-\-2.%0-|-1| .
Note 3.) and becaufe in this Cafe —<= is
the laft Term to be multiplied into V 2ay—y,
fem ES
therefore ABD = vy -|- - 2 V aay |; y7 x
mii y mm Ae ee |
Example Xt.
U _ ; ! I
Let z= a becaufe in this Cafe r= —,
a
m == 1, therefore the Term laft breaking off
Ze Aa hese
aah Ato Ay cilia * , whence g=4 4,
nxn-|-2xn-|-1| ~ %
and 4 z is the laft Term to be multiplied by
V 2ay-\-9?, therefore | |
ABD
iM — Cori Q 5 7
A aa : “Sa
ABD = : oa aa yx ra
ce E+“. And if you take y= V3 aa,
there will arife a Geometrically Quadrable-
Area rea fitting — oe Abfciffe, viz. Area = 2,
i a ea a - x aa V0? :
a I have shes Theorems of this Nature,
for Figures depending on the Circle and Pa-
rabola; but thefe two may fuffice as a Spe-
cimen to fhew the Ufe of my Method pub-
lif’'d in my Treatife of Quadwatures, in de-
termining the Quadratures” of Irrational Fi-
guges, for which there has been no Method
(as far as I know) as yet made Publick.
That the Reader may the more eafily
come at the Invention of thefe and the like
Theorems, I fhall fubjoin another, and more
hereafter, ‘if need be. |
- Let therefore (Fg. 40.) ACE be! a Se-
micircle, ADE a Geometrically Irrational
‘Curve, whofe Ordinate BD cuts the Semi-
circle in C. Let the Quantities be denoted
as before, wiz. the Diametér AF = 2a, the
Abfcifle AB = y, the Arc AG = 4: the Or-
dinate BD =<} and let z = rv 27, an E-
quation exprefling the Nature of the Curves
‘ADE, in which r denotes any given and de-
termin’d Quantity, and # an indefinite Ex-
‘ponent of the indetermin’d Quantity xT
fay the Area’ *
| $ ABD
258 Mifcellanea Curiofa.
242-\-1 , i.
- ABD = fe qu* -\-v V sayy? %
Leal |
——2 : Paes “|-
n-\-1 | nkn-|-1|
AA, itl yy aby by ere T
ma yt 2 Sie ——___—+ me ed 3 =.
career I——-2,
ACK 2-5 aDKX2n—7 :
ins POT 4.| yt - 54
(gan Oe eas
AEXUM—9 yb 9. ie
mere MNS
m-\-11
ara R2n|T yg a? AK IH—T
n~a1 fd
gl RAPES OAT 390 <8 Cae ae
mt xn-\-t| FBS
2 2f,. tT gs
A” BY LMm—3 op ae ik Beh) eee ee
eG eat 7 Sy a eae
on Ome?
;
In ‘this Theorent thefe Things are to be
taken notice of; C1.) That ’tis. made up of
two Infinite -Series,. the former of which
(connected by the Sign -|-) is.multiplied in-
to v V sayy? ; but the Terms_of-the latter
(affeGted, by the Sign.—) are Abfolute. (2.)
_ That in the. former Series, the Capital. Let-
ters, A, B, C, @c. denote the Coefficients of
the Terms refpectively preceding them, and
m the latter have the famg Values as in the
is * Tae ’ former
ie load Curiofa. 2 : 9
Merci (3-) That the Quadrature may be
exprefs’d by a finite Quantity, when » is a
pofitive Integer, or equal to nothing, or if
22 be an odd Number; for in thefe Cafes
each Series is broke off. (4.) That 29 is
equal to the laft Term breaking off, of the
former Series.
Example I.
uv” : a
het =. Becaufe in this Cafe x =o,
a ae i.
Fis s, therefore fhall the Area ABD =
v?-|-2Y V ray—y?—24).
COROLLARY.
The witole Figure AFE is equat to twice
the Square, whofe fide is ACF, lefs the
Square of the Diameter.
Example A.
Let rl becaufe id this cae n= T;
i et deste ‘al dhe! Area ‘ABD = =
pen
ie
blvow. | i e yu?
460 Mifcellunea Curiofa.
ual Pe ae aA Bo i y ae Ly?
aa" ta” — LT Seria
ey,
a*
Example Il.
yo Ne ‘ .
Letz =>, becaufe in this Cafe # = 2,
I
a>
c=
B 2
, therefore fhall the Area ABD ="
=
Nr ners:
a ye ee eee
ies
oy) STs ee
274. 2.18 3
While I was writing this, I receiv’d the
late Months of the Lipfick A&s, mn which L
read, with a deal of pleafure, feveral excel-
lent things for promoting Geometry; and
among them fome Remarks of Mr. Lezbuitz,
and Mr. ¥. Bernouili, upon my Method of
Quadratures. - In the its of April, An. 1695.
Mr. Lecbnitz informs us that he has a Me-
thod fomewhat like ours; and truly, I migh-
- tily Congratulate my. felf, that any thing of
mine could have the leaft likenefs to the
Thoughts of fo great-a Geometer. But
whereas he fays his own is much more Ge-
neral, and fhorter than mine; I make no
doubt of that. It were to be wifh’d, he
ee ge would
— MifceHanea Curiofa. 261
would no longer fupprefs this Method of
his, and feveral other things he has, efpe-
cially relating to his Differential Calculus, but
rather, as foon as his Leifure permits, pub-
lifh them for the Good of the Common-
wealth of Learning. In the mean while we
hope the Illuftrious Marquefs De PHofpital
will fpeedily make publick what is neceflary
to perfe& that Calculus, in the latter part
of that excellent Work of his, which Gn
the Preface to the former part) he informs —
us, he has compos’d upon the Integral Caleu-
lus. We expedct.alfo, with fome Impatience,
that other Section, in which that Noble
Author promifes he will fhew the Ufe of
his Calculus in Phyficks and Adechanicks. For
whatever he has publifh’d, as well thofe
Specimen to be found featter’d in the Lip-
fick Aéts, and elfewhere, as that excellent
Book of his CIntitul’d, <Axaly/e des Infiniment
petits) caufe us to expeé great Things from
that Noble Marquefs. ae
Whereas the Ingenious Mr. . Bernouslls has
thought fit Gin the Acts of. February and Au-
guft, An. 4695.) to fay my Method is not
General, I freely confefs it, as that Saga-
cious Perfon might eafily perceive in the
Courfe of my Examples. In a Matter fo
Intricate I took what Steps 1 could; and if
deter’d with the length and difficulty of the
Journey, I then made no farther Progrefs.
I might fairly make a Step where I pleafe,
fince my Application to thefe Mathemati-
cal Studies,is only by the by. Mr. Bernouiiti
has partly hinted where my Method is at a
sot 3.3 Stand,
262 Mifcellanea Curtofa.
Stand, though he feems not to have taken i
up the whole Matter. In the mean while —
J acknowledge my felf ‘highly oblig’d, that —
he has honour’d my Treatife with his Ani-
madverfions; but much more fo, that he -
was willing to free me of my Miftakes,
with fo much Candor and Humanity. :
ore ie
a“
Do a
ca eR tng tC
4 : -
i
Ba oy ager ie in Re
2 ES ame
Ss eee oe
— Zoos
he)
a raen
M ifcellanea Curifa. 262.
Concerning the apparent Magnitude
fs of the Sun and Moon, or the ape
rent diftance of two Stars when
mgh the Horizon, aad when
higher elevated. |
| if Do not defign fo much to eftablifh any —
_ thing of my own that may be fatisfactory
in folving this admirable appearance, as to
_ dete& the Errors of thofe that have offered
at a Solution thereof, and have come fhort
(as I conceive) of being fatisfa@tory ; that
thereby I may again fet the minds of Philo-
fophers on work, and roufe them up to en-
quire anew after this furprizing Phexomenon.
That I may do this the more effectually, I
_fhall briefly declare the Matter of FaQ, and
_ then proceed to the Reafon thereof, given by
-feveral, and to their Confutations.
Firft therefore itis well known that the
mean apparent Magnitude of the Moon is
30 m. 30 f. we will take it Numero Rotundo
to de 30, that is, an Arch of a great Circle
in the Heavens of 30 Minuts 1s covered by
her Diameter; and this we'll fuppofe ta be
her apparent Diameter, at a full Moon in the
midft of Winter, and when fhe’s in the Me-
ridian, and at her greateft Northern Lati- _
tude, and confequently the utmoft that fhe -
can be elevated in our Horizon; ’Tis as well
bird | eS a Known
264 Mafcellanea Curiofa.
known alfo that when fhe is in this pofture,
being looked upon by the naked Eye fhe ap-
_ pears (that we may accommodate all to fen-
fible Meafures) to be Magnitudims Pedalis,
about a foot broad. But the fame Moon be-
ing looked upon juft as fhe rifes, fhe appears
to be three or four foot broad, and yet if
with an Inftrument we take her Diameter,
both in one pofture and t’other, we hall find
that ftill fhe fhall be but. 30 Minutes; the
feveral ways of trying this I will not mention,
they being as various as are the Methods of.
taking the Moons apparent Diameter, com-
mon enough among the Aftronomers , neither
will I infift upon the truth of the Matter of.
Fad, for that I think cannot reafonably be
queftioned, after fo many trials and fo many
experiments thereof, faithfully recorded by
undoubted Witnefles; and it would be very
unreafonable to imagine that fo many Au-
thors fhould rack their Brains for folving an
appearance wherein they were not certain
of the matter of FaG@. But becaufe of Wul-
“lius in Verba, ¥ can affert that I have ac-
curately try’d it my felf, and I have fo
found it: One of the ways I proceeded was
thus; I took avery good Telefcope of about
6 foot long, in the inward Focus of whofe
Bye-Glafs | apply’d a very fine Lattice made
of the fingle hairs of a Man’s Head; then
looking with this at the Moon when fhe was
juft rifen and looked extraordinary big, I ob-
ferved what number of the fquares of the
Lattice were occupy’d by her Body ; then ob-
ferving her again, when more elévated and’
free from all extravagant Greatnefs, I ftill
bE £ yee = ' Faia * 3 ee A eh r: found
Mifcellanea Curiofa. 265
found the fame fquares of the Lattice pof-
- feffed by her. This way is equivalent to that
now more ufed, of taking her Diameter by
Mr. Townly’s Micrometers, but I have alfo
tried and found the fame thing by an accu-
rate Sextant, taking the diftance of the
Moons oppofite Limbs. ;
Now this Phenomenon affords two things to
be confidered, firft why the Moon (I ftill
name the Moon as being an Objeé& more
adapted for our fight , for the fame thing
holds in the Sun) fhould feem bigger about
the Horizon, then when more elevated ; and
-_ fecondly, fhe appearing bigger, how comes it.
to pafs that her Diameter being taken, it is
no greater than when fhe appears lefs. But
the Difquifition concerning this latter being
likely to comprehend the former, | fhall not
divide my Difcourfe into two Branches, but
proceed in the Method propofed. Only I
defire it may be noted, that I fuppofe-the
Horizontal and Meridional Moon to be found
both of the fame Angle, whereas in truth the
the .Meridional Moon (tho’ appearing lefs)
fhall be found of the greater Angle: which
increafeth the Wonder. But this proceeding
from the different diftances that one and the
other is looked at (the Meridional Moon be-
ing nigher us by almoft a Semidiameter of the
Earth) and confequently eafily folved that
way; I havetherefore chofen to put hetween
them a plain equality, for avoiding Coufufion
and Intricacy in Difcourfe.
| Wherefore let us hear what the Ingenious
_ of thefe latter days can fay to this appearance.
And firft we find the Celebrated Des-Cartes at-
oo tributing
266 Maifcellanea Curiofa.
tributing this appearance rather to a deceived
Judgment than to any Natural Affe@tion of
the Organ or Medium of fence; forthe Moon
(fays he) being nigh the Horizon, we have a
better opportunity and advantage of making
an Eftimate of her, by comparing her with
the various Objects that incur the fight, in its
way towards‘her; fo that tho’ we imagine
fhe looks bigger yet ’tis a meer deceit; for
we only think fo, becaufe fhe feems nigher
the tops of Trees or Chimneys or Honfes or
a fpace of Ground, te which we can compare
her, and eftimate her thereby ; but when we
bring her to the Teft of an Inftrument that
cannot be deluded or impofed updn by thefe
appearances, then we find our Eftimate
wrong, and our Senfes deceived. Thefe
Thoughts, methinks, are much below the ac-
cuftomed accuracy of the noble Des Cartes ;
for certainly if it be fo, I may at any time in-
-creafe the apparent bignefs of the Moon, tho’
in the Meridian; for it would be only by
getting behind a Cluiter of Chimneys, a Ridg
o: a Hill, or the top of Houfes, and compa-~
ring, her to them in that pofture, as well as
in the Horizon, befides if the Moon be look’d
at juft as fhe is Rifing from an Horizon de-
termined by a fmooth Sea, and which has no
more Variety of Objects to compare her to,
than the pure Air ; yet fhe will feem big, as
if lookt at over the rugged top Of an uneven
Town or recky Country. Moreover, all va-
riety of adjoining Objects may be taken off, ~
by lookiag through an empty Tube, and yet
the deluded imagination ts not at ‘all helped
thereby. Icome next to the folution hereof
: Mee | given
Mifcellanea Curiofa. 267
given by the famous Thomas Hobbs, and for
this we fhall ftand in need of Figure at.
wherein, fays he, let the point G be the Cen-
ter of the Earth, and F the Eye on the fur-
- face of the Earth ; on the fame Center G let
‘there be ftruck the two Arches, EH deter-
mining the Atmofphere, and A D to repre-
fent that blue furface in which we imagine
the fixed Stars; and let FD be the Horizon.
‘Divide the Arch A D into three equal parts
by the lines BF, C F, it is manifeft that the
Angle AFB is greater than the Angle BFC,
and this again greater than the Angle CFD.
‘Wherefore fays he, to make the Angle CFD
equal tothe Angle CPD, the Arch CD mu
be greater than the Arch CB ; and confe-
quently, that the Moon may in the Horizon
appear under the fame Angle as when ele-
vated, fhe muft cover a greater Arch, and
therefore feem greater; that is, the Moon
in the Meridian appearing under the Angie
BFC, that fhe may appear under an equal
Angle in the Horizon, as fuppofe CFD, ’tis
neceflary the Arch CD fhould be greater than
CB; and confequently tho’ fhe appear to ful
tend a greater Arch when in the Aorizox
then when elevated, yet fhe appears under
the fame Angle. And all this without Re-
fraction. The Geometry of this Figure is
moft certainly true and demonftrable. At
this I quarrel not ; Lut it makes no morein
our prefent Difficulty than if nothing had
been faid ; for the Philofopher has here made
a Figure of his own, and from thence he ar-
gues as confidently, as if Nature would ac-
commodgte her felf to his Scheme, and he
a A net
265 Mifcellanea Curiofa.
not oblig’d to accommodate his Scheme te
Nature; for here he has made the Circle GF
reprefenting the Earth very large in propor-
tion to the Circle AD; and then indeed
taking the point F in the Harth’s furface,
and by lines from thence dividing the Angle
AFD into what ever equal parts the inter-
cepted Arches AB, BG, CD, fhall be un-
equal. But if he had» confidered, that the —
Earth is as it were a point in refpect of the
Sphere of the fix’d Stars, nay the very annual
Orbit of the Earth is almoft if not altogether
imperceptible (faving the truth of Mr. Aook?s
Attempt) he would have found that the Lines
FB, FC, FD, mutt be allconceived as drawn
from the point G, and then equal Angles ©
will intercept equal Arches, and equal Arches
equal Angles: And fo it happens (at leaft
beyond the poffibility of difcovery of fenfe)
to the Eye onthe furface of the Earth. And
befides he fhould have confidered , that all
Obfervations Aftronomical are performed as
from the Center of the Earth, and therefore
it is that they keep fucha ftir about a Paral-
lax; fo that his drawing his lines fo far from
Gas F is, and to, another concentrick Circle
fo nigh as AD, deceived him in this
Point. | oe |
The famous Gaffendus has written 4 large
Epiftles on this Subject, the fubftance of all
which is, that the Moon being nigh the Ho-
vizon and looked at through a more foggy ~
Air, cafts a weaker Light, and confequently
forces not the Eye fo much as when brighter;
and therefore the Pupil does more inlarge it
fclf, thereby tranfmitting a larger Projection -
tis on
TSS eevee a, el a ee
Mifcellanea Curiofa. 269
on the Retiza. In this Opinion I do find he
is not alone, for in the Journals des Scavans
this Difquifition being again revived by a
French Abbe, he therein follows this Senti-
ment of Gaffendus. It was firft publifhed in the
3d Conference prefented to the Dauphin in
August 1672. but by reafon of an Objection
moved by Father Pardye, it was fain to be
re-publifhed with fome additions and amend-
ments in Oétob. 1672. The addition was,
that this contracting-and enlarging of the
Pupil caufeth a different fhape in the Eye;
an open Pupil making the Cry/taline flatter
and the Eye longer, and the narrower Pupil
fhortning the Eye, and making the Cry/talline
more convex, the firft attends our looking at
Obje&ts which are remote or which we think
fo; the latter accompanies the viewing Ob-
jects nigh at end. Likewife an open Pupil
and flat Cryftalline attends Obje&ts of a more
fedate Light, whilft Obje&s of more forcible
Rays require a greater Convexity and narrow
Pupil. From thefe Pofitions the Abbe endea-
-voured to give an account of our Phenomenon
as follows. When the Moon is nigh the
Horizon, by comparifon with interpofed Ob-
jects, we are apt to imagine her much farther
‘from us then when more elevated, and there-
fore (fays he) we order our Eyes as for view-
ing an Object farther from us ; that is, we
fomethigg enlarge the Pupil, and. thereby
_ make the Cryftalline more flat; moreover the
duskifhnefs of the Moon in that pofture does
not fo much ftrain the fight; and confe-
quently the Pupil will be more large, and the
‘Cryftalline more flat: Hence a larger Image
fhali
270 . Mifcellanea Curiofe.
thall be projected on the Fund of the Eye,
and therefore the Moon fhall appear larger.
And this difpofition of the Eye that magnifies
her, magnifies alfo the -divifions of our fore-
mentioned Lattice, and confequently the by
her Body fhall poflefs no more of the divifions
than when fhe feems. lefs. Thefe two fore-—
mentioned accidents, viz. the Moons ima-
ginary diftance and duskifhnefs, gradually
vanifhing as fhe rifes,a different Speczes'is here-:
by introduced in the Eye, and confequently
fhe feems gradually lefs and lefs, ’till again
fhe approaches nigh the Horizon. Thefe two
Opinions of Gaffendus and the Abbe being fo
nigh a-kin , I fhall confider them both toge-
ther, and firft I affert that a wider or nar-
rower Aperture increafes not, neither dimi-
nifhes the projection on the Retina. I know
Honor atus Faber in his Syzopfts Optica endeavours
to prove the clear contrary. to thismy Affer-
tion, and that after this manner. Fig. 42.
' AB isan Objet, E F the greater aperture
of the Pupil, adniitting the projection KI
on the Retina, whereas the lefler aperture -
CD admits only the projection GH; but
GH is lefs than KI, ,wherefore,a leffer aper-
ture diminifhes the projection. 1. admire
that any Man that. undertook (as Honoratus
Faber) to write of Opticks more accurately
than all that went before him, fhould. be
guilty of fo very grofs.an Error; and,I. do
more admire that the celebrated Gaffendus,
-and with-him.the noble; Hevelius fhould be. of |
the fame Opinion: For tho’ the *forefaid Fi-
gure and Demontftration hold moft. certainly
true in diredt proje@ions, as in_a.dark Room
2 2 : ‘ with
Maifcetanea Curtofa. 274
with a plain’ hole; yet it will not hold in
Projections made by RefraGtion, as it is in
thofe on the Retiva in the Eye, by means of
the Cry/tallime and other Coats and Humours
of the Hye. Fora Demonftration of this ob-
ferve Fig. 43. wherein let A B be a remote
Object, and BF the Cryflalline at its large
aperture, projecting the Image 1M_ on the.
Retina. Let then CD be the lefler aperture
of the Pupil before the Cry/ftaline : 1 fay the
Image IM fhall be projected as large as be-
fore, for the Cone of Rays EAF confifts
_ partty of the Cone of Rays CAD, therefore
where the former EAF is projected, the
latter CAD, as being a part of the former,
fhall be projected alfo. So that no more is
effected by this narrow Aperture , but that
the fides of the radiating Cones are inter-
cepted, .and confequently the Point I fhall be
affected. with lefs light,, but it fhall. ftill be in
the fame place: What is faid of that Cone
and that Point may be faid of all other Cones
and other Points of the Obje&. From:hence
appears firft, the Invalidity of the Account
given of. the Moons appearance by Gaffendus
from this Reafon. 2dly, The Reafon ap-
pears why a Telefcopes greater or lefler
Aperture, makes no difference in the Angle
it receives; for imagine EF to be an Objedct-
glafs of a Telefcope, and ’tis plain. 3dly,
’Tis. evident. why a greater or lefs Aperture
on a.Telefcope fhonld make the Objects ap-
pear Lighter or Darker, for thereby more or
lefs Rays are admitted to determine on the
Projection of each Point. But all this by the
by. And this is fuficient fora Bearnenion.
| | OQ
«
542 : Mifcellanea Caria
of Gaffendus and Faber : But our forementioned
Abbe fuperadds to a greater or leffer Aper-
ture of the Pupil, as a neceflary Confequent,
a greater and leffer Convexity of the Cry/tal-
line, as alfo a lengthening and fhortening the
Tube of the Eye. And this 1 muft confefs
would do fomething if we find it truein our
Cafe; and this let us try. Firft, fays he,
the duskifhnefs of the Moon nigh ‘the Hori-
zon admits the Pupil to enlarge it felf, the —
Cryfrailine to flatten, and the Eye to lengthen.
But what if we change our Objed, and in-
ftead of the Moon take the diftance between
fome of the fixt Stars; as fuppofe thofe of
Orions Girdle) we fhall ‘ind the fame Pheno-
menon in them, and yet I hope neither he nor ©
Gaffendus will aflert, that they at one time
ftrain the Eye more than at another, or that
at any time their fulgur ftrains the Eye at
all; if he do, let him take Stars of the lefler
Magnitudes , nay even thofe that can but
juft be perceived, and then he will be tof
vinced: Or let him confider whether this will
hold in looking at the Sun through very dark
Glaffes, which render the Sight thereof as
inoffenfive to the Eye; as that of a green.
“Field. But perhaps he will then fay that this
other Reafon holds, which is 2dly, That the
preater imaginary diftance at which we think
the Moon near the Horizon, than when more ©
elevated, makes us Contemplate her as if
really fhe was fo, viz. with ample Pupils, @c.
but this I have fuficiently overthrown in my
Remarks againft Des Cartes; therefore I pafs
it over, only fubjoining that if there were
any thing in this Surmife, my-thinks the A-
rizontal
Mifcellanea Curiofa. 272
rizontal Moon fhould be fancied nigher to us
than farther from us; for if we are for try-
ing natural Thoughts; let us take Children
to determine the Matttr, who are apt to
think that could they go to the edge of that
{pace that bounds their Sight , they fhould
be able (as they call it) to touch the Sky ;
and confequently the Moon feems then rather
nigher to us than farther from us.
After I had writ thus. far I accidentally caft
my Eye upon Riccioli’s Treatife of Refraction,
at the end of his 24. Volume of the Almageft,
Lib. 10. Sett. 6. Cap.1. Queft. 13. wherin he
fpeaks of our prefent Difficulty ; buf to my
wonder I find him affert, that he and Father
Grimaldi had often taken the Horizontal Sunand
Moons Diameter by a Sextant, when to the
naked Eye they appeared very large ; (Grimal-
dus directing his Sight to the left edge, and
Ricciolus to the right,) and that even by the
Inftrument they always found the Diameters
greater than when more elevated, the Sun
often fubtending an Angle of almoft a De-
gree, and frequently 45 Minutes, the Moon
alfo 38 or 40 Minutes. This is downright
contrary to the matter of Fa which I have
before alledged, and diretly repugnant to
the matter of Fad aflerted by the French.
Abbe in the forecited Journal. Whether of
tis be in the right I leave to accurate Ex-
periment to determine, and fubmit the |
whole to the decifion of the IMu/frious Roy-
al Society. Only give me leave to add
one word againft Riccioli, for had his Experi-
ments been accurately profecuted, he fhould
fig have
274 Mifcellanea Curiofa.
have tryed them when the Horizontal Moon
- had look’d ten times more large in Diame-
ter than ordinary; and then if it’ be true,
that even by an Inftrument fhe will be found
proportionally broader than really fhe fhould
fubtend an Angle of 300 Minutes, or 5 De-
grees: for very often I have feen the Moon
when fhe appeared 10 times broader than or-
dinary, which the fmall addition of 8 or to
Minutes to. her ufual Diameter will never
Caufe. }
~ Laftly,as an Apology for my reviving this
difquifition to that Noble Company of Exglifh
Philofophers, I fhall only imitate the words
of the forementioned Abbe’s Letter. Pour la
Raifon de cette Apparence, & de la tromperie de
nos Sens, je la tiens plus Difficile a trouver, que
les plus grands Equations d Algebre, & quand
vous y aurex bien penfe, vous m’ Obligerez de m
en dire voftre Sentement, &c. PS ode
After which | have only to fubfcribe my
felf-an unworthy Member, and an humble
Servant and Admirer of that Iluftrious Com-
pany. :
machi. 83, William Molyneux.
The
M. ifcelanea Curiofa. 2 7 5
The Sentiments of the Reverend
and Learned Dr. John Wallis
_R. S. Soc. upon the aforefaid
Appearance, communicated in a
Letter to the Publifber.
A S to the laft Inquiry (concerning which,
you fay, the Royal Society would be
glad to know my Opinion ;) about the appa-
rent Magnitude of the Sun near the Horizoz,
greater than when confiderably high :
The Inquiryis Ancient : And, I remember,
I difcourfed it near forty Years ago with
Mr. Foffer, then Profeffor of Aftronomy in
Grefham College. Who did then affure me
(from his own Obfervation, I fuppofe, for
I have never examined it my felf,) that the
apparent Magnitude taken by Inftrument
(however the Fancy may apprehend it) is not
greater at the Horizon, than when higher.
And Mr. Cafwel (when your Letter was com-
municated to our company here) affirmed the
fame. _ oes |
And (though I have not my felf made the
Obfervation) I do not doubt but the thing is
fo. For it is agreed, That Refraction near
the Horizon, though (as to appearance) it
alter the Altitude of the thing feen ; yet it
alters not the Azimuth at all. )
se And
276 Mifcellanea Curiofa.
And it muft needs be fo. For, fince this
equally refpects all points of the Horizon 5 let
the Refraction be what it will, the whole Ho-
rizon can be but a Circle: So that there is —
no room for the breadth of a thing (as to
the Angle at the Eye) to be made greater,
whatever its Tallnefs may (the Refraction
not equally affecting all parts in the Circles
‘of Altitude.) Nor is there any reafon why
this fhould rather thruft the other, than that
the other thruft this, out of place. , :
Whereas, in the Altitude, it is otherwife :
For while what is near the Horizon is inlarg-
ed, that which is further off is thereby con-
~ traé&ted: which as to the Azimuth or Horizon-
‘tal Pofition cannot be. —
In Speétacles indeed it is otherwife ; for
they reprefent the Object every way enlarg-
ed; and do thereby hide the adjacent parts.
But in Refraction by Vapours, fuppofing all
parts of the Horizon equally affected by them,
one part cannot be expanded in breadth
(whatever it may be as to the heighth) with-
out thrufting out another (for the whole
Horizon can be but a Circle) and, why one
part rather than another ? Peis
Unlefs we would fay (as perhaps we may,
if there fhall appear a neceflity for it) That
the Rays of a lucid Body do expand them-
felves every way to the prejudice of the parts
adjacent, by covering them. weet
But fuppofing (which I am apt to believe,
till the contrary fhall be evinced by Experi-
ment) that the Sun or Moon’s apparent Dia-
meter taken by Inftrument near the Horizon,
is the fame as taken in a higher Pofition, (I
: 3 mean
Mifcelanea Curiofa. 279
mean its Horizontal Diameter, or that paral-
Jel to the Horizon ; forthe erect Diameter, in
a Circle Perpendicular to the Horizon, may by
the Refraction be varied, and thereby made,
not greater, but lefs than when higher; as
hath been noted in the Name of Sol Elkpticus
at the Horizon.) Suppofing, I fay, that
the Sun’s apparent Diameter Horizontal, ta-
ken by Inftrument, is the fame near the Ho-
rizon, asin a higher Pofition, I take its Ima-
ginary greatnefs which is fanfied near Horizon,
to be only a deception of the Eye ; or rather
the Imagination from the Eye. :
For fure it is, that the Imagination doth
not eftimate the greatnefs of the Objed feen,
only by the Angle which it makes at the Eye;
but, by this compared with the fuppofed di-
ftance. |
True it is that, Ceteris paribus, we judge
that to be the greater Obje&t, which makes
at the Eye the greater Angle: But not fo if
apprehended at different Diftances.
For if through a Cafement (or lefler aper-
ture) we fee a Houfe at 1oc Yards diftance 3;
this Houfe (though feen under a lefs Angel)
doth not to us feem lefs than the Cafement
through which we fee it, (or this greater than -
that, becaufe it makes at the Bye the greater
Angle:) But the Imagination makes a com-
parative Eftimate from the Angle and Di-
{tance jointly confidered. :
_ So that, if two things feen under the fame
or equal Angles, if to one of them there be
ought which gives the apprehenfion of a
greater Diftance, that to the Imagination will
appear greater. |
bios Fi3 2 Now
278 Mifcellanea Curiofa.
Now fure it is,that one great advantage for
Eftimating of a ‘thing feen, is, from the va-
riety of intermediate Objects between the
_ Eye and the thing feen. For then the Ima-
gination muft allow room for all thefe
things.
Hence it is that if we fee a thing over two
Hills, between which there lies a great Valley
unfeen, it will appear much nearer than if
we fee the Valley alfo: And it will appear as
juft beyond the firft Hill. And if we move
forward to the top of the neareft Hill (that
fo the Valley may be feen) it will then appear
much further than before it did. —
And on this account it is, that the Sun
fetting, appears to us as if it were but juft
beyond the utmoft of our vifible Horizon ; -be-
caufe all between that and the Sun is not
feen. And,upon the fameaccount, the Hea-
ven it felf (ceineiConmgataE ty the vifible
fTorizon.
Now when the Sun or Moon is, near the
Horizon, there isa profpe& of Hills, and Val-
lies and Plains and Woods, and.Rivers; and
variety | of Fields, and Inclofures, between
it.and us’: which prefent to our Imagination
a great Diftance capable of) receiving all
thefe:» Or,:if it fo chance that (in fome Po-
fition) thefe Intermediates are not adually.
féen :: Yet having beemaccuftomedto fee them,
the Memory fuggelts to:us a view as large, as
is the-vifible ddorix0n. |
' Bat/!wiren® the Sum or; Moon j isin a higher
Pofition; we fee nothing between us andthem
<unlefs perhaps fome Clouds), and:therefore
nothing to prefent to our Imagination fo, gt eat
ve Diftance as the other j Be “And
a Mifcelanea Currofa. 279
_ And therefore, though both be feen under
the fame Angle, they do not appear (to the
Imagination) of the fame bignefs, becaufe
~ not both fanfied at the fame Diftances: But
that near the Horizon is judged bigger (be-
caufe fuppofed farther off) than the fame
when ata greater Altitude.
?Tis true, that as to fmall and middling
_Diftances (befides this Eftimate from Inter-
mediates) the Eye hath a means within it felf
to make fome Eftimate of the Diftance. As,
when we already know the bignefs of a thing
feen, to which we have been accuftomed ; as
a Man, a Tree, a Houfe or the like: If fuch'
thing appear to us under a fmall Angle, and
-indifting, and faintly coloured; the Imagi-
nation doth allow fuch Diftance, as to make
fuch a thing fo to appear. And, if this, thro’
a Profpective Glafs, be repefented to us un-
der a bigger Angle, and more diftiné: It is
accordingly apprehended as fo much nearer. -
But the cafe is otherwife, when we do not
by the known bignefs, judge the Diftance;
but, by the fuppofed Diftance, judge of the
bignefs ; asin the Cafe before us. A
_ And accordingly, different Perfons, accord-
ing to. different fancied Diftances, judge ve-
ry differently. As, if two Stars be fhewed
to ignorant Perfons, and you ask how far they
feem to be afunder: one perhaps will fay a
Foot; another. a Yard, or more: And one
fhall fay, the Sun appears to him as big as a
_Bufhel ; another, as big as a Holland Cheele :
pe eitimating according to the fancied Di-
‘ftance. beat v
T ‘4 c. Again
280 Mifcellanea Curiofa.
Again ; in our two Eyes (when the Object ©
is feen by both) there is yet. another means —
of eftimating how far off it is. (And'it is |
this by which we judge of Diftances.) Name- —
ly, there are, from the fame Obje&, two dif-
ferent vifual Cones, terminated at the two
Eyes: Whofe two Axes contain, at the Ob- ©
jet, different Angles, according to different
Diftances: An accuter Angle at a great —
Diftance, and more obtufe when nearer.
Now, that fuch Obje& may be feen by both
Eyes, clearly; it is requifite that the Eyes be
put into fuch a Pofition, as that the Sight of
_ each Eye receive the refpective Axe at right
Angles. Which requires a different Pofition
of the two Eyes, according to the different
Diftance of the Objet. | a
As will manifeftly appear ; ifwe look, with
attention, on a Finger (or other fmall Ob-
ject) at two or three Inches diftance from the
Kye; and then upon another like Obje& at
three or four ¥ards beyond it: (and this al-
ternately feveral times. For *twill be mani- ©
feft, that while we look intently on the one, ©
we do not fee the other (or but confufedly}
though both be juft before us. And, as we —
change our view, from the one to the other, |
we manifeftly feel a Motion of the Eyes (by
sab Mufcles) from one pofture to ano-
hon BR to gi wh vag hs 2 |
And according to the different pofture in —
the Eyes, requifite to a clear Vifion by both,
we eftimate the Diftance of the Object from —
pS: | a
ve And hence it is,that they who have loft the
_ Bight of one Eye, are at a great difadyantage,
ay 5 Filia \ Pe iy MT RR rae as
Mifcellanea Curtofa. 28:
as to eftimating Diftances, from what they
could do while they had the ufeof both. _
But now when the Diftance grows {o great,
as that the Pofition of thefe vifual Axes be-
come Parallel, or fo near to Parallel, as not
to be diftinguifhable from it: This advan-
- tage is loft, and we can thenceforth only
conclude, that it is far off; but not how far.
Hence it is, that our view can make no
- diftin@tion of the Moon’s Diftance, from that
of the other Planets, or even of the fixed
Stars: But they feem to us as equally remote
from us ; though we otherwife know their
-Diftances from us to be vaftly different. Be-
caufe the Parallax (as I may fo call it) from
the different Pofition of the two Eyes, isquite
loft, and undifcernable, in Diftances much
lefS than the leaft of thefe.
_ And fo, of the fixed Stars among them-
felves : Which, though they feem equally re-
mote from us; many (for ought we know)
be at Diftances vaftly different. Nor can we
tell, which of them isneareft: (unlefs perhaps
we may reafonably guefs, thofe to be neareft,
which feem biggeft.) Becaufe, here not on-
ly the Parallax from the Diftance of the two
Eyes 3 and that from the Earths Semidiame-
ter ; but even that from the Semidiameter of
the Earths great Orb, is quite loft ; and none
remaining, whereby to eftimate their Diftance
from us. i | |
_ But (to return to our cafe in hand ;) tho’
as to fmall Diftances, we may make fome
eftimate from the known Magnitude of the
Obje& : And, as to middling diftances, from
the Parallax (as I may call it) arifing from
Pi the
282 Muifcellanea Curiofa.
the interval of the two Eyes; Yet even this
latter will hardly reach beyond, if fo far as
the vifible Horizon: and all beyond it, is loft.
So that, there being nothing left to affift
the fancy in eftimating fo great a diftance,
‘but only the- intermediate Objects : Where
thefe intermediates appear to the Eye, (as,
when the Sun or Moon are near the Horizoz -)
the diftance is fancied greater, than where
they appear not, (as when farther from it :),
and confequently (though both: under the
fame or equal Angles) that near the Horizon
is fancied the greater. And this I judge to
be the true reafon of that appearance.
You will excufe ([hope) what excurfion I
have made; becaufe though fome of them
might have been fpared, as to the prefent
- eafe; yet they are not impertinent to the bu-
finefs of Vifion ; and the eftimate to be thence —
made, of JAdagnitudes and Diftances, by the
Imagination. | | at Ae
The Sun’s Eclipfe Aéay 1/f. was here ob-
ferved about 4 a Digit ; between one and two
a Clock after noon. be
~ Mifcellanea Curiofa. : 282
A Demonfiration of an Error com-
mitted by common Surveyors in
comparing of Surveys taken at
long Intervals of Time arifing
from the Variation of the Mag-
netick Needle, dy William Mo-
lyneux. E/; “Eb BSe oa
HE Variation of the Magnetick Needle
is fo commonly known, that I need not
infil much on the Explication thereof; ’tis
certain that the true Solar Meridian, and the
Meridian fhewn by an Needle, agree but in
a very few places of the World; and this too,
but for a little time (if a moment) together.
The Difference between the true Meridian
and Magnetick Meridian perpetually vary-
ing and changing in all Places and at all
Times; fometimes to the Eaftward, and
fometimes tothe Weftward.
On which account ’tis impoffible to com-
pare two Surveys of the fame place, taken at
diftant times, by, Magnetick Iniftruments,
(fach as the Circumferentor, by which. the
Down Survey, or Sir William Petty’ s. Survey .of
Ireland was taken) without due allowance be
made for this Variation. To which purpofe
we ought to know the Difference between.
the Magnetick Meridian and true Meridian
at
284 Mifcellanea Curiofa.
at that time of the Dowz Survey, and the
faid difference at the time when we make a
new Survey to compare with the Dowz
Survey. _
But here I would not be underftood as if
I propofed hereby to fhew, that a Map of
the fame place, taken by Magnetick Inftru-
ments at never fo diftant times, fhould not at
one time give the fame Figure and Comtents as _
at another time. This certainly it will do
moft exactly, the variation of the Needle
having nothing to do either in the Shape or
Contents of the Survey. All that is affected
thereby, is, the Bearings of the Lines run by
the Chatn, and the Boundaries between
Neighbours. And how this may caufe a con-
fiderable Error (Cunlefs due allowance be ~
made for it) is what I fhall prove moft
fully.
In order to which, let us fuppofe that about.
the Year 1657. (at which time the Dowz
Survey was taken) the Magnetick Meridian
and true Meridian did agree at Dubliz, or
pretty nigh all over Jreland ; that is to fay,
that there was no Variation. And indeed
by Experiment it was at that time found, as
I am well aflur’d, that at Dublin it was hardly
half a Degree. |
Let us fuppofe that in the year, 1695. the
Variation was 7 Degrees from the North to
the Weftward ; that it was really fo, I be-
lieve Iam pretty well aflured , from an Ex-
periment thereof made by my ielf with all
diligence. But this is not material, let us ©
now only fuppofe it. HAL aed
Let
Mifcellanea Curiofa. 285
Let AB reprefent the Survey of two
Town-Lands, one in the pofleflion of 4, and.
vother in the pofleflion of B, which we call
A Town-Land and B Town-Land, taken by
the Down-Survey , Anno 1657. when there
was no Variation.
Let the Line NS running through the
Point P be the true Meridian, and confe-
quently the Magnetick Meridian alfo at that
time, becaufe of the fuppofed no Variation,
and let this Line NS be alfo the Boundary
between the two Town-Lands 4 and B.
In the year 1695. when the Variation ‘is
7 Degrees from the North to the Weftward,
B having a Map of the Down Survey, and be-
- ing fufpicious that his Neighbour -4 had in-
croached on him by a Ditch PQ, imploys a
Surveyor to inquire into the matter: The
Surveyor finds by his 4¢@ap that the Boundary
between B and his Neighbour 4 run from the
Point P through a Meadow dire@ly according
to the Magnetick Meridian SPN; but ob-
ferving the Ditch P © caft up much to the
Eaftward of the prefent Magnetick Meri-
dian, he concludes that 4 has incroached on
B, and that the Ditch ought to have been
caft upalongft the Line Pq, the Angle Q Pq
being an Angle of 7 Degrees , that is is the
piclent Variation of the Needle; and the
ine Pq the prefent Magnetick Meridian:
For which Variation , not making any al-
lowance, he pofitively determines that B has
all the Land in the Triangle Q P 4, more
than he ought to have; and that his Ditch
ought to run alongtt the Line P q.
Bas
286 © Mifcellanea Curiofa.
’Tis true indeed, if the Surveyor go the
whole furround of the Lands Aand B, he
will find their Figure and Contents exactly
agreeable to the Map here exprefled. But
then the Bearings of the Lines are all 7 De-
grees different from the Bearings in the Map,
and they will run in and out upon the adja-
cent Neighbouring Lands, and caufe endlefs
Differences between their Poffeffors; as is
manifeft from the Figure: wherein the prickt _
Lines reprefent the Difagreement in the
Bearings of the Lines, protracted from the
Point P'; and we fee A incroaching on his
Neighbours on the Weftward, as he incroaches
on 8B, and Bs EHaftward "Neighbours in-
croaching on him, and fo forward and clear
round. Whereas, by a due allowance for the
Variation of the Needle, all this Confufion
and Difagreement is avoided, and every
thing hits right.
Thus for inftance in the Cafe before us,
knowing that the Magnetick Variation has
caufed the prefent Magnetick Meridian to
fallin the Line z g P s, 7 Degrees from the -
‘North to the \Weftward ; ; to reduce this to
the Magnetick Meridian at the time of the
Down Survey, I muft make the Meridian of
my Map to fall 7 Degrees to the Eaftward of
my Magnetick Meridian ; as we fee the Line
PQ falls 7 Degrees to the Eaftward of the
Line P q:
What is here faid on fuppofition that the
Magnet had no Variation at the time of the
firft Survey taken, and that it had 7 De-
grees variation Weftward:at the time of the
fecond Survey, may ealily He accommodated
Ta
Mifcelunea Curiofa. 287
to the fuppofal of any other Variations at
the firft and fecond Surveys, Afutatis mue
tandw, for knowing the Variations we know
_ their Difference; and if we know their Dif-
ference, this gives us the Angle O. Pq, by
which we reduce them to each other.
The beft way therefore to make Maps in-
variable, conftant and everlafting , were for
the Surveyors , who ufe Megnetick Inftru-
ments to make always allowance for the
Magnetick Variation, and to protraé& and
lay dew their Plats by the true Meridian.
This a wary Sailer is fully convinced of :
and therefore in Steering his Courfe, he con-
{tantly allows for the prefent Variation, which
he obferves by the Azuuth Compafs, or elfe
he would mifs his appointed Harbour oftner
then he would hit it: For ‘ two Points on
the Globe keep the fame Bearing to each
other by the Magnetick Meryidian for any
time together: And though, the Variation
be flow, yet in a long Courfe, or in times
pretty diftant, 1t may caufe vaft Errors, un-
lefs allowed for. Thus for inftance, fup-
pofe in the year 1660. a Sailor had fteered
from the Land’s end of Exgland to Cape Fini-
‘fter in Spain, by his Magnetick Compafs
a direct South Courfe; and that at thattime ~
there were no Variation. Afterwards Azzo
1700. when there was (fuppofe) 7 Degrees
of Variation from the North to the Weft-
ward, another Sailor intending to make the
fame Paflage, fteers diretly the fame South-
erly Courfe by his Magnetick Compafs: I fay, .
this laft Seaman will be carried far into the
Bay of Bifcay to the Haftward, and will ier
| 5
\
288 Mifcellanea Curiofa.
of his defired Port by many Leagues ; but if
in his Courfe he hath allowed for this Varia-
tion, and inftead of failing a direé Southerly
Courfe by his Compafs, he had fteer’d 7 De-
grees from the South to the Weftward, he
had hit his Point. Whether thefe be the true
Bearings of thefe two Places, it matters not :
_ We go onto the Suppofition that they are.
Perhaps it may be objected, That Surveys
may be taken withoutMagnetick Inftruments,
and that therefore this Error arifing from the
Megnetick Variation, and Change of the
Bearings of Lines, may be avoided. To
which I anfwer, firft, That granting a Sur- >
vey may be taken without Magnetick Inftru-
ments, this is nothing againft what we have
laid down relating to Surveys that are taken
with Magnetick Inftruments, as the Down
Survey actually was, and moft Surveys at pre-
fent aétually are taken therewith. Secondly,
Though a Survey may be taken truly with-
out Megnetick Inftruments, fo as to fhew the
exact Angles and Lines of the Plat, and con-
fequently the true Contents, yet this will
not give the true Bearings of the Lines, or
fhew my Pofition in relation to my Neigh-
bours, or the other parts of the Country. |
This muft be fupply’d by the Magnet, or,
fomething equivalent thereto, as finding a
true Meridian Line on your Land by Celeftial
Obfervations. And I doubt not but the an-
cient Egyptians, before the difcovery of the
_ Magnet were forced to fome fuch Expedient
in their Surveys and Applotments of Lands
between Neighbour and Neighbour, after
the Inundations of the Nile, which, we are
| - told
Muifcelianea Curiofa. 289
told, gave the firft Original to Geometry
and Surveying. Abfolute Neceflity and Ufe
having introduced thefe, as Delight and Di-
verfion introduced Aftronomy amongft the
Chaldeans. |
- And this brings me to another Objection
which may be made againft the Inftance be-
fore laid down: It may be faid, That cer- |
tainly the Surveyor which B imployed was
very ignorant, who would choofe to judge
of the Line P QO, rather by its bearing than
by determining the. Point QO, by meafuring
from AH and G. Tothis I anfwer, What if
both the Points HandG were vanifh’d fincethe
Down Survey was taken ? What if the whole
face of the Country were chang’d, fave only
the Point P? and the Line P O? How fhall
the Surveyor then judge of the Line P O but
by its bearing ? That this is no extrava-
gant Suppofition, we have an Example in
Egypt above-mentioned, where the Nile lays
all flat before it, and fo uniformly covers all
with Mud, that there is no diftincion. In
fuch a Cafe your bearing muft certainly help:
you out, there is no other way. |
But I anfwer fecondly, To fay that the
Surveyor might have determin’d the Point Q
by admeafurement from G and #, or any
other adjoining noted Points, as from F, kK, /,
&e. tis very true; But then ’tis again{t our
Suppofition. Iam upon fhewing an Error
that arifes from judging of the Line P O by
Magnetick, bearing, and to tell me that this
‘might be avoided by another way, is to fay
nothing. I my felf oe it may be hae |
. e
290° «Mifcellanea Curiofa. _
ed by allowing for the Variation; but ftill it
is an Error, till it be avoided. NC
But thirdly, if B’s Surveyor do not allow
for the Variation of the Needle, he will ne-
ver exactly determin even the Points G, F,
H, K, &c. orany other Points in the Plat;
but inftead thereof will fall on the Points g, /,
” From what has been laid down, we may
fee the abfolute neceflity of allowing for the
Variation of the Magnet, in comparing old
Surveys with new ones; for want of which
great Difputes may arife between neighbour-
ing Proprietors of Lands: and it were to be
wifh’d that our Honourable and Learned
_ . Judges would take this Matter into their
Confideration whenever any Bufinefs of this
kind comes before them. Hitherto an abfo-
lute Acquiefcence in the Down Survey, with-
out any of the fore-mention’d Allowance, has
been agreed upon as a ftanding Rule in our
Courts of Judicature in Ireland, but that ma-
ny Men may be injured thereby, I fuppofe
is manifeft from what foregoes. fom THE
I have only this to add, That leaft I be
thought herein to ftrike at the Truth or Ex-
actnefs of the Down Survey, ’tis not at all
the intention of this Paper, but rather to
confirm it, by fhewing which way Men
ought to Examine it truly, and not by the
common ways ufed by them, which rather
confound it, and all that claim under it. |
See the Table Fig. 44.
Mifcellanea Curiofa. 29%
Although this Paper was chiefly defigned
for the ending of Contefts in the Kingdom of
4reland about the interefts of fome of thofe
whofe Lands are Neighbouring, and have
been furveyed by Magnetick Inftruments,
yet confidering its univerfal Ufe, it was
thought it would be very grateful to the Cu-
tious to publifh it here. :
292 Maifcellanea Curiofa.
A Propofal concerning the Parallass
of the fixed Stars, m Reference
- tothe Earths Annual Orb. In”
feveral Letters of May the 2d.
June 29. and July 20.1693. from
Dr. John Wallis to William
Molineux Ef; 73 |
Sig. :
| Am obliged to you for two Books which
you have been pleafed to fend me, that
of your Sciothericum Telefcopicum, and that of
Dioptricks;, which you have performed fo well,
that Ihave not been better fatisfied with any
that I have feen of that Subje@. J fhould not
fo long have neglected to return my Thanks
for them, but that I thought a Letter of bare,
Thanks to be too empty, unlefs I had fome-
what elfe to fend with it. - oo
You will, 1 hope, give me leave (though I
have not the opportunity of being perfonally
known to you) to fuggeft a Speculation, which
hath beenin my Thoughts thefe Forty Years
or more ; but I have not had the opportu-
nity of reducing it to Practice, as being not
fo well ftored with neceflary Inftruments of
that kind, nor much exercifed to Telefcopick
Obfervations. And though I have many
Years fince fuggefted it to others, yet nel-
Bee ther
— Mifcellanea Curifa. 293
ther have they had leifure of convenience of
putting it in Practice. : : :
It is concerning the Parallax of the fixed
Stars, as tothe Earths Annual Orb. |
Galileo complains of it a great while fince
Cin his Syffema Cofmicum) as a thing not at-
tempted to be obferved with fuch diligence
as he could wifh, and I doubt we have the
fame caufe of. complaining ftill. I know that
Dr. Hook and Mr. Flamftead have attempted
fomewhat that way, but have defifted before
they came to.any thing of Certainty. What
hath been done to that purpofe abroad I
know not. | |
Galileo hath fuggefted divers things confi-
derableinordertoit. — 7
As to the times of. Obfervation ; That it
fhould be when the Sun or Earth are in the
Tropicks, or as near thereto as may be: Be-
caufe at thofe times, if any, will be the
greateft difference obfervable in their meri-
dional Altitude. :
As to the Stars to be obferved, That they
fhould be fuch as are as near as may be to
the Pole of the Ecliptick: For fuch as are in
the. Plain of the Ecliptick, or near unto it,
though they may be fometime nearer, fome-
time farther from us, (which might fome-
what alter their apparent Magnitude, if it
were fo, much as to be obfervable) yet it
would little or nothing alter the Parallac-
tick Angle, as Galilelo doth there demon-
iteate..
He notes alfo, that in a bufinefs fo nice, °
the ordinary Inftruments of Obfervation
(though pretty ee would be infafficient
vt
3 (ne
294 Maifcellanea Curiofa.
(he doubts) for this purpofe, and doth pro-
pofe, that by the fide of fome Edifice or Moun-
tain, at fome Miles diftance, the fetting of
fome noted Star (as that of Lucida Lyre)
might be obferved at thofe different times of
the Year, which might be equivalent to an
Inftrument whofe Radius were fo large.
_ Which were a good Expedient if Practica-
ble; but I doubt the Denfity of our Atmo-
fphere is fo great, as that it will be hard to
difcern a Star juft at the Horizon, or even
within fome few Degrees of it: And that the
Refraction would be there fo great, and fo
uncertain, as not to comply with fo curious
an Obfervation. | Ap nH
That which occurred to my Thoughts upon
thefe Confiderations, was to this purpofe ;
That fome Circumpolar Stars (nearer to the
Pole of the Equator than is your Zenith, and
not far from the Pole of the Zodiack) fhould
be made choice of for this purpofe. And in
cafe the Meridinal Altitude be difcernably
different at different times, fo will alfo-be
their utmoft Eaft and Weft Azimuth, which
may be better obferved than their Rifing or
Setting: And this will be not obnoxious to
the Refra&ion, as is the Meridional Alti-
tude ; (for though the Refraction do affect the
Altitude, yet net the Azimuth at all); and
we may here have choice of Stars for the pur-
pofe; which in Obfervations from the bot-
tom of a Well we cannot have; being there
confined to thofe only which pafs very near
our Zenith, though very fimall Stars.
Maifcelanea Curiofa. 295
I would then take it for granted, as a thing
_ at leaft very probable, that the fixed Stars
are not at all (as was wont to be fuppofed) at
_ the fame diftance from us; but the diftance
of fome, vaftly greater than of others; and
confequently, though as to the more remote,
the Parallax may be undifcernable; it may
‘perhaps be difcernable in thofe that are near-
er to us.
And thofe we may reafonably guefs (tho’
We are not fure of it) to be neareft to us,
- which to us do appear biggeft and brighteft,
as are thofe of the Firft and Second Magni-
tude; and there are at leaft of the Second
Magnitude, pretty many not far from: the
Pole of the Ecliptick, (as that in particular,
in the Shoulder of the lefler Bear): And in
-cafe we fail in one, we may try again and a- |
gain on fome other, which may chance to
_-be nearer to us than what we try firft. And
Stars of this bignefs may be difcerned by a
moderate Telefcope, even in the day-time;
efpecially when we know juft where to look
for them.
The manner of Obfervation I conceive,
may be thus: Having firft pitched upon the
‘Star we mean to obferve, and having then
sconfidered (which is not hard to do) where
-fuch Star is'to be feen in its greateft Eaft or -
Welt Azimuth ; it may be then convenient
to fix! very firm-and fteadily on fome Tower,
‘Steeple, or other high Edifice (in a convent-
-ent fituation) a good Telefcopick Objedt-glafs
in fuch pofition, as,may be proper for view-
ing that Star. And-at a due diftance from it
near the Ground, build on purpofe (if alrea-
U 4 dy
296 Mifcellanea Curiofa..
dy there be not any) fome little Stone Wall,
or like Place, on which to fix the Bye-glafs,
fo as to anfwer that Object-glafs: And ha-
ving fo adjufted’ it, as through both to fee
that Star initsdefired Station, (which may beft
be done while the Star is to be feen by Night
in fich fituation, near the time of one of the
Solftices), let it be there fixed fo firmly, as
not to be difturbed, (and the place fo fe-
cured, as that none come to diforder it),
and care be taken fo to defend both the
Glaffes, as not to be endangered by Wind
and Weather. In which contrivance I am
beholden to Mr. Yohu Cafwel M. A. of Hart-
ball in Oxford, for his Advice and Affift-
ance ; with whom Lhave many Years fince,
communicated the whole matter. —
This Glafs being once fixed (and a Mi-
crometer fitted to it, fo as to have its Threds
perpendicular to the Horizon, to avoid a-
ny inconvenience which might arife from
diverfity of Refra@ion if any be) the Star
may ‘then be viewed from time to time (for
the following Year or longer) to fee if any
change of Azimuth can be’ obferved.
This I thought fit to recommend to your
Confideration, who’ do fo well underftand
Telefcopes, and the managery of them ; not
knowing any who is more likely to’ reduce
it to Pra@tice. Ff you fhall: think fit to give
your felf the trouble of attempting the Ex-
periment, and ‘that it fucceed well, it’ will
be a noble Obfervation, » ‘and worth ‘the ‘La-
bour : And, if it fhould‘mifcarry; ‘the “imei:
| Hope would not be ef Pade
” “fut
Mifcellanen Curtofa. 3 297
But when I fuggeft (asa convenient Star
‘for this purpofe) the fhoulder of the lefler
’ Bear (as being the neareft to the Pole of the
Zodiack of any Star that is of the firft or
fecond Magnitude), I do not confine you to
that Star; but (without retracting that)
fuggeft another ; namely, the middle Star,
in the Tail of the great Bear, which (tho’
-‘fomewhat farther from the Pole of the Zo-
diack) isa brighter Star than the other, and
may be nearer to us. ,
But I do it principally upon this Confider-
ation : namely, That there is adhering to it
a very fmall Star, (which the Arabs call
Alcor, of which they have a Proverbial fay-
ing, when they would defcribe a fharp-fight-
‘ed Man; That he can difcern the Rider
on the middle Horfe of the Wayn, and of
one who pretends to fee fimall things but o-
ver-look much: greater , Vidit Alcor at non
Lunam plenam): Which Hevelius in his Ob-
fervations, finds’to be diftant from it.about
- 9 Minutes, and 5 or 10 Seconds: Sothat -
befides the advantage of difcovering the Pa-
rallax of the greater Star, if difcernable.
‘Their difference of Parallax of that and
of the lefler Star“(being both within the
reachiof a Micrometer) may do our Work .
as well. For if that of the greater Star be
difcernable, but that of the leffer be either
motidifeernable; or Jefs difcernable.. Their
different idiftances ‘from each other at dif-
ferent times of the the Year, may, perhaps
(without farther: Apparatas) be difcerned by
‘a good»Telefcope of a competent length,
| tur nifhed with a Micrometer, if carefully
: Fhe pre-
998) = =Mazfcellanea Curiofa.
_preferved from being difordered in the Inter-
vals of the Obfervations; and difcover at
once, both, that there 1s a Parallax, and
that the fixed Stars are at different diftan-
ces from us, wherein, that I be not mifta-
ken, my meaning is not, that the Inftrument —
or Micrometer fhould be removed for the
obferving of the lefler Star; but that (when
the Azimuth of the greater Star is taken)
by a Micrometer (confifting of divers fine
Threads parallel and tranfverfe) may -(at the
fame time) be obferved the Diftance of the —
two Stars, each from other, in that Pofiti-
on (both being at once within the reach of
the Micrometer ;) which diftance (the In-
ftrument remaining unmoved) if it be found
(at different times of the Year) not to be
the fame ; this will prove, that there is a —
diflerent Parallax of thefe two Stars. me
This latter part of the Obfervation (of
their different diftances at different times)
J fuggeft, as more eafily practicable though
not fo nice asthe former. For it may bedone
I think, without any further pparatas there
than a good Telefcope, of ordinary form,
farnifhed with a Micrometer, (this being
carefully kept unvaried’ during the Interval
of thefe Obfervations.. And if this part only
of the Obfervation (without the other) be
urfued 3 it matters not though the two Ob-
Fopineiiag (near the two Solftices) be, one at
the Haftern, the other at the Weftern Azi-
muth (whereby both may be taken in the .
Night-time,) for the diftance muft (at both —
Azimuths) be the fame, if after obferving
the Azimuth of the greater Star it be me |
é CEie
Mifcellanea Curiofa. 299
ceflary to move the Micrometer for meafur-
ing its diftance from Alcor that may be done
another Night (and it is not neceflary to be
done atone Obfervation) for that diftance,
and cannot be difcernably varied in a Night.
or two. aS ify
I fhall give you no farther trouble at pre-
fent, but fubfcribe my felf, Sir,
Yours, &c.
A
300 Mifcelanen Curiofa. —
A Difcourfe on this PROBLEM;
Why Bodies diffolved in Menfirua
Specifically lighter than them-—
felves, frou therems ~~
By Mr. WiLLiaM MotyNeux, of
Dublin, Member of the Royal Soczety, —
“FA4HE Liberty of Philofophifing being
now univerfally granted between all
Men, I am fure that a difference in Opinion
will be no breach of affeGtion between two in-
tirely Loving Brothers: And therefore I fhall —
take the freedom’to propofe my own Thoughts
in a matter wherein my Brother Mr. The-
mas Molyneux hath appeared publickly in the
Novelles de la Republique des Letres, Mois @
Aout 1684. Art 4: and Afou de Fanvier 1685.
Art 7. The Problem propofed is, Why Bodies
diffolved float in Liquors lighter than themfelves ?
as for Example: Mercury diflolved in ftrong
Spirit. of Nitre fwims therein, tho’ each fmall
Particle of Mercury, be far heavier than fo
much of the Liquor whofe place it occupies.
This, fays he, cannot be folved by the prime
Law of Hydroftaticks, which is, that a Bo-.
dy which is an equal quantity is heavier than
a like quantity of Liquor, finks in that Li- ® —
: : quor ;
“
- Maifcellanea Curiofa. 201
‘quor ;: thus a Cubick Inch of Iron being hea-
vier than a Cubick Inch of Agua-Fortis, and
each Particle (how fmall foever) of Iron be-
- ing heavier than a like Particle of Agqua-Fortis 5
Iron being put into iAgua-Fortis fhould fink,
and yet we find, that Iron being diffolved in
a convenient quantity of Agua-Fortis floats
therein, and does not fall to the Bottom.
The Reafon which my Brother gives for this
is, That the internal Motion of the Parts of
the Liquor, does keep up the Particles of the
diffolved Solid, for they being fo every Mi-
nute, are movable by the leaft force imagi-
nable, and the AGion of the Particles of the
Menftruum, is fuiicient to drive the Atomes
of the diffolved folid Body from place to
place; and confequently, notwithftanding
their Gravity, they do not fink in the Liquor
lighter than themfelves. Asa Proof of this
in the 7th Article of Fanvier 1685. he offers
an Experiment known in Chymiftry, that a
_ Menftruum over a digefting Fire (as the Chy-
mift {fpeaks) will diffolve a greater quantity
of a Body put into it, than when ’tis off the
Fire, andif it be taken offthe Fire, and fuf-
fered to cool, a great Portion will precipi-
tate of that which was perfe&ly diflolved,
whilft the Afenftruum continued hot. | For,
fays be, the Particles pf the ALenfruum ac-
quire a more violent agitation by the Fire,
and are therefore able to raife and. keep up
a greater Quantity of the diflolved Body, or
hereby they are able to refift a greater Gra-
vity. |
it has been objected againit this Notion,
that the common Experiment of precipita-
tion,
—
202 Maifcellanea Curwfa. -
tion, by mixing an Alkaly with an Acid feems —
to contradi& this; for thereby the Fluidity |
of the Aenftruum is not taken away, and
confequently, the internal Agitation of its
Parts is not diminifhed, and yet thereupon,
the Particles of the diffolved Body precipi-
tateall to the Bottom. ‘To this he anfwers
in the forecited Article of Fanuary, that all
Mixtures of different Liquors introduce in
each a different Conformation of Pores, and >
therefore the Infufion of a new Liquor, drives
the infefible Parts of the diffolved Body from
their Places, and forces them to ftrike a-
gainft each other, and cling together, and —
fo becoming more big aad heavier tha. for-
merly, the internal Agitation of the Liquor
is no longer able to move and fuftain them,
and confequently theyfall to the Bottom.
This, as fairly and fhortly as I can propofe
it, is his Sentiment of this Phenomenon.
But I conceive an other.-Account may be
given of this Appearance, and that the fore- —
faid Law of Aydroftaticks is alittle deficient. -
Tis true indeed, if we confider only the fpe--
cifick Gravity of a Liquor, and the fpecifick
Gravity of a folid Particle floating therein,
the forementioned Rule is exact; but in
finking there is requifite a feparation of the
Parts of the Liquor by the finking Body; and
there being a natural Inclination in the Parts
of all Liquors to Union arifing from an A-
greement or Congruity of their Parts, there
isa refiftance therein to any thing that fe-
parates this Conjunction: Now unlefs a Bo-
dy have weight enough to overcome this
Congruity or Union of Parts, fuch a Body
will
_ Maifcellanea Curiofa. — 303
will float in a Liquor fpecifically lighter than
it felf. But that a heavy Body, as Adercury
or /ron may. have its Parts reduced to that
Minutenefs, that their Gravity or Tendency
downwards, is not ftrong enough to feparate
'the Cohefion or Union of the Parts of a Li-
quor, will be manifeft, if we confider, that
the Refiftance made by the Afedium to a fal-
- ling Body, is according to the Superficies of
the Body; but asthe Body decreafes in Bulk,
its Superficies does not proportionably de-
creafe, thus a Sphere of an Inch Diameter,
- has not eight times lefs Superficies than a
Sphere of two Inches Diameter, tho’ it have
eight times lefs Bulk, and confequently paf-
fing through a Adedivm, as fuppofe Air or
~ Water, the Sphere of an Inch Diameter is,
proportionably to its Bulk, more refifted, than
a Sphere of two Inches Diameter in propor-
tion to its Bulk, and hence it will come to
-pafs, that at laft a Body may be reduced to
that Minutenefs, that its Gravity prefling
downwards (which is according to its Bulk)
may be lefs than the refiftance of the Aé¢edi-
‘um, which operates on the Surface of the
Body; feeing as I faid before, the Surfaces
of Bodies do not decreafe fo faft as their
Bulks, thefe decreafing in a Triplicate, but
—thofe in a Duplicate Ratio,of the Bodies Dia-
- meters.
This Account does not at all oppofe the
Experiment of a Menftruum over the Fire,
being able ta diflolve or fuftain a greater
Quantity of a heavy Body; for the Reafon,
of this, as’tis given by my Brother, does not
Contradi¢t my Notion. The Account nes
: wife
ie -: Mifjedlamee Curiofel
wife, that He gives of Chymical Precipita- |
tion agrees very well with what} propofe :
So that of thefe I fhall fay no more. :
But becaufe in the beginning of my Dif
courfe, I fay that the forementioned: Law of
_ Hydroftaticks isa little defective, 1 defire toex-.
plain my felf a little further in that Point. In
Weights falling through the Air, were Gra-
vity only confider’d, the Proportions of their
Defcents would be exadtly as Galileo has de-
monftrated ; but it is allow’d by-all, that the
Refiftance of the Air, not being. confider’d in
thofe Demonftrations, they are not Mathe-
matically true in Praétife, but that really
there is fomething of that proportion hind-
red by the Airs Refiftance. Now, what is
this lefs than to fay, that the Refiftance of
the Air takes off fome of the Operatien of
Gravity, or is able to withftand or oppofe
part of its AGion? And if fo, what fhall we
fay were an Iron Sphere let through a Me-
dium of Water? Surely the Proportions of
its defcents would be much more difturbed
herein, as Water is much more Solid and dif-
ficult to be feparated or pafled through than
Air; and confequently we muft needs grant,
that more of the Operation of Gravity, is
_ taken off or refifted by this Oppofition of the
Water, than that of the Air. And. if fo,.
furely there may be a certain degree of Gra-
vity,; that may be quite taken off by the
_refiftance of the Water: Were a Piltol Bul-
Jet let fall through the Air, it would defcend
imperceptibly nigh the Propor tions that Ga-
fileo has afligned, but were a fingle grain
of Sand fo let fall, it would-be much hiadred
in
”
“3
i -
i 44
Maifcellanea Curiofa. 205
in its Courfe, and the half of this Grain would
be more obftructed ; what fhall we then fay ~
of the ten thoufandth part, or of a part the
ten thoufand millioneth of this, and agai of
the Infinits Subdivifions of that, till at laft
we come to a part that would be wholly re-
Sifted, or kept up; fuch as I conceive the
‘minute Particles of a Body diffolved in a
Menftruum ? tae es
- On this account ’tis, 1 fay, that the fore-
mentioned Principle of Aydrofraticks is a little
defective; for it confiders not the natural’
Congruity of the Parts of a Liquor, whereby
they defire, as *twere, to unite and keep to=-
gether, juft as we fee two drops of Water on ~
‘adry Board being brought together do jump
and coalefce, and therefore Liquors have an
innate power of refifting a certain degree of
force that would feparate them; fuch as I
fuppofe the degree of Gravity, in the moft
minute Particles of a Body diffolved ina
Menftruum. 7 ;
_ The fore-mentioned Rule holds true to the
moft nice Senfe in great Bodies, but in thofe
that are by many Millions of Divifions {mal-
ler, it feems to fail. he |
“This, in fhort, is my Conjefture in this
matter, which I propofe, as my Brother did
his, with all fubmiffion imaginable, and there
by to give occafion to others to enquire into
‘the Caufes of this appearance, rather than to
publifh my own Sentiments as the undoubted
Solution thereof. eee
But this 1 muft acknowledge, that the in-
ternal motion of the parts of a Liquor feems
fo very agreeable to truth, and explicates fo
| ° 4 many
306 = Mifcellanea Curiofa.
many Phenomena eafily and plainly, that I
would not be thought to deny it. Neither
would I be thought wholly to reject my Bro-
thers Solution of this Problem 3 for certainly
that Motion (whatfoever it is) in a A/en-
flruum, which is able to diffolve fuch a folid
‘Body as Iron, that is, which is able to di-
fturb the clofe and ftrong Cohefion of the
Parts of Iron, may very well be fuppofed fuf-
ficient to difturb or keep up thefe parts from
refting in the bottom of the Veffel, wherein
the folution was made: And certainly no
better account can poflibly be given of fuch
Solutions, than by fuppofing fuch an internal
motion in the parts of the Adenftruwm infinu-
ating themfelves into the folid Body, and
loofening its parts. And tho’ it may be ob-.
jected, that in the parts of Water there may
be fuppofed as violent an internal motion, as
in the parts of Aqua-Fortis, and yet we fee
Water will not diffolve Iron as Aqua-Forte
does, and common Bees-Wax is difturbed by
neither of them, I leave the nice enquiry
after this Point to others, wiz. What kind of
Motion and peculiar Conformation of parts
is requifite both in the Adenftruum and in the
diffolved Body, that a Solution may refult
from their Commixture. Rieti dt ,
Some Reflections on the foresoime
Paperby, dio kee ata
What my Brother has laid- down in this
Difcourfe,; [think does moft undeniably evince
that the received Law of Alydroffaticks is
‘Aomewhat detective. For Liquors, tho’ they
fl ia * r are
| Mifcellanea Curiofa. 307
are Fluid yet they are Bodies, and therefore
confift of parts united 3 which Union, tho’ it
be eafily deftroy’d, yet of neceflity it requires
fome degree'of Force for the efleGting it; nor
is it more manifeft, if rightly confidered,
that a Flint requires Force for the feparation
ef its parts, than that Fluids do for theirs.
But however, I imagine, this Property ought
not to be relied upon as the fole Caufe of this
appearance, to which my Brother has apply’d. |
it; nay perhaps does not fo muchas concur
the leaft in the producing this effect; my
Reafon in fhort is this: Whatever is of fuff-
cient Power to raife the minute Particles of
a heavy Body in a light Fluid, is certainly a
fufficient caufe to keep them in that ftate :
Now my Suppofition may give fome account
of this, what my Brother fays, never can 3
for he muft neceflarily fuppofe them firft
raifed ; and then he gives the reafon of their
not finking: Whereas ’tis not to be quefti-
oned but that that Force which raifed them,
is the fame which keeps them from falling to
the bottom.
But thefe Conjefures (for I efteem them
no more)! leave to the Confideration of thofe
that defire to enquire further in this Mat~
eer.
¢
1 ag? Mnifcellanea sir |
of Of the aveig ht of a ole foot of er
vers grams, &c. try’ "ama Vef-
fel of well-feafon’'d Oak, whofe
concave eons an cnt cape foot.
By the direction of the Philofo-
phical Society at Oxford. |
HE following Bodies were poured
gently into the Veffel, and thofe in the
12 firft Experiments were weighed in {cales
turning with 2 ounces, but the laft 7 were
weigh’d in feales turning with one ounce.
The pounds and ounces here mentioned ‘are
Avoirdupois.
, 7 aa, a ae
1. A foot of Wheat (worth 6s.a -
Bufhel ) weigh’d of Avoirdupois_
weight. 47. 8
ae ers of the beft fort (worth
4.4. a Bufhel) 43. de
3. “Che huh fort of Wheat meafured .
a fecond time. Oo. ae
Both forts were red Lammas Wheat of 4
the laft year.
4. White Oats of the laft year. 29. ©
The beft fort of Oats were 2d. ina
Bubhel better than thefe.
5- Blue Peafe (of the laft year) and
much worm-eaten. | 49. 1%
6. White Pea/e of the laft year but one. 50. %.
q. Barley
. Mifcellanea Curtofa. 309
| ib. 3.
», Barley of the laft year (the beft
fort fells for 1s. 6d. ina Quarter
more than this. : Als 2.
8. Malt of the laft years Barley, made
2 Months before. 130. % 4.
9. Field-Beans of the laft year but one. 50. 8.
10. Wheaten AZeal (unfifted). St. Ge
11. Rye Aeal (unfifted.) wgB. fe
12. Pump Water. 62. 3%.
13- Bay Salt. 34. 1
14. White Sea-Salt. 43: 2.
rg. iSand.. 85. ge
16. Newcaftle Coal. G7. 2.
17. Pit-Coal from Wednesbury, 63:
but this is very uncertain in the
filling the Interftices betwixt the
“greater pieces. 63. 0.
18. Gravel. | 109. §.
19. Wood-Afhes. 3 58. 5.
A further Lift of the Specifick Gravities of
Bodies, being in proportion as the following —
numbers.
ee. 7000
Fir dry 546
Elm dry 600
Gedar dry Ol
Walnut-tree dry - 631
Crab-tree meanly dry |
Ah meanly dry, and of the out-fide lax
part of the Tree 734.
Afhmore dry, but about the Heart 845
Maple-tree e855
: X 3 Yew
7645
2 10 Mifcellanea Curiofa.
“Yew of a Knot or Root 16 years old 760
Beech meanly dry W854
Oak very dry, almoft Worm eaten 133
Oak of the out-fide fappy Betts fell’d a
year fince 870
Oak dry, but of a very found ‘clofe tex-
LUEE 2 — 929
The fame tried another time 932
Logwood SEO OLS
Claret : nye id eu QOS
Moil Cyder not clear | 1017
Sea-water fetled clear He Ose
College plain Ale the fame : 1028
Urine | 1030
Milk | t aedt. DORE
Box the fame ~ . 1031
Redwood the fame es LOSE:
sack Jh2tO33
Beer Vinegar vey 1034
Pitch i J115@
Pit-Coal of Stafford- fhite : ¥240
Speckled wood of Virginia Stee G
Lignum Vite — pL a27
Stone-bottle C7 Lae
‘Ivory : . 1826
Alabafter : 1872
Brick | ° 1979
Heddington-ftone, . foft lax kind. 2029
Burford-ftone, an old dry piece *") ¥)g0a8)
Paving-ftone a hard fort from about |
Blaidon t 2460
Flint 7 2542
Glafs of a quart Bottle, > 2666
Black Italian Marble 2704
Ww hite Italian Marble tried twice . 2707
White
—— MifeeNanea Curiofu. = 311
White Italian Marble of another fort
of a vifibly clofer texture 2718
ppiock-tin re / 7321
Copper ae : $843
Lead ye 11345
Quick-filver 14019
Quick-filver 2 ee es
The laft Experiment was tried with another
quantity of Quick-filver, which had been
ufed in Water in the preceding Experiment:
However I rather truft the laft, for that I
found a fimall miftake (tho’ here in the cal-
culation allawed for) in the weight of the
Glafs containing the Quick-filver in the trial
before. a
The Solids here mentioned were exami-
ned hydroffatically by weighing them in Air
and Water; but the Fluids, by weighing an
equal portion of each ina Glafs holding about
a quart. The numbers fhew the proportion
of gravity of equal portions of thefe Bodies ;
but if of thefe Bodies we take portions
equally heavy, their magnitudes will be reci-
procally proportional to their correfpondent
numbers, e,g.a cubic foot of water isto a cubic
foot of Alabafter in gravity as 1000 to 1872;
but a pound weight of water, is toa ponnd
weight Albafter in magnitude as 1872 to 1000.
So that knowing by the former Table the
weight of acubic foot of Water, and by this,
the proportion in gravity betwixt Water
and Alabafter, we may by the Rule of Three
find the weight of a cubic foot of Alabafter,
and fo of any other of thefe Bodies ; or we
‘may know their magnitude by knowing their
X 4 : gla~
giz Mifcellanea Curiofa.
gravity. Sothat an irregular piece or quan-
tity of thefe Bodies being offered, *tis but
weighing them, and we may know their juft |
magnitude without further trouble.
Obfervations of the Comparative, Intenfive
or Specific Gravities of various Bodces.
Made by Mr. J. C. ; |
Ump-water, — Ag 1000
Cork, ce fe 2.37
Saflafras Wood, it ‘ 482
Juniper Wood (dry) ~ ‘> §56
Plum-tree, (dry) 663
Maftic, : 849
Santalum Citrinum, : 809
Santalum album, 1041
Santalumrubrum, 1128
Ebony, : L177
Lignaum Rhodium, - yo eae
Lignum Afphaltum, : 1179
Aloes, | 1177
Succinum pellucidum, oes (Ge
Succinum pingue, : 1087
Jet, 1238
The top part of a Rhinocero’s horn, 1242
pp iT, +2
‘The top part of an Ox horn, os us a OAC
The (Blade) bone of an Ox, \ +S
An human Calculus, 1240
Another Calculus humanus, me
Another Calculus, ° 1664,
Brimftone, fuch as commonly fold, 1815
Borax, — Wigs te 172Q:
A fpotted faGtitious Marble, 1822
& Gally-Pot,) 1 tain : 1929
Lue sa Oylter-
Mifcellanea Curiofa.
Opter-ihell,,
Murex-fhell,
Lapis manati,
Selenitis,
- Wood petrefied in Lough-Neagh,
Onyx-ftone,
-Turcois-ftone,
Englifh Agat
Grammatias lapis,
A Cornelian, ©
_Corallachates,
Tale.
Coral,
Hyacinth (fpurious)
-Jafper ({purious)
A pellucid Pibble,
Rock Crytftal,
Cryftallum Difdiaclafticum
A red Patfte, :
Lapis Nephriticus,
Lapis Amiantus from Wales,
Lapis Lazuli 7
An Hone,
Sardachates,
A Granat,
A Golden Marcaiite,
9
_ Ablue Slate with fhining Particles,
A mineral Stone, yielding 1 part in
160 Metal,
The Metal thence extra¢ted,
The (reputed) Silver Ore of Wales,
The Metal thence extracted,
Bifmuth, |
Spelter, .
Spelter Soder,
Tron of a Key,
es}
2092
2590
2290
2322
2345
2510
2508
aSlz
2515
- 2568
2605
2657
2689
‘2635
2666
264%
2659
2704.
“42842
2894
2913
3054.
3288
3598
3978
4589
3500
2650
8500
7404
LIO87
9859
7O6§
3362
7943
Steel,
Steel,
Caft Brafs,
Wrought Brafs,
Hammer’d Brafs,
A falfe Guinea,
A true Guinea,
Sterling Silver,
A brafs Half-Crown,
Ble4trum, a Britifh Coin,
A Gold Coin of Barbary,
A Gold Medal from Morocco,
A Mentz Gold Ducat,
A Gold Coin of Alexanders,
A Gold Medal of Queen Mary,
A Gold Medal of Queen Elizabeth,
214. Mifcellanea Curiofa.
7852
“8100
8280 |
3349
9075
18888
10535
9468 —
12071
17548
18420
18261
bhoreys | ue
19100 |
19125
A Medal efteem’d to be near fine Gold, 19636 |
— Mifcelanea Curiofa. 315
o 2 HERO Laa
yee! *
A Letter of Dr. Wallis to Dr.
Sloane, concerning the Generation
of Hail, and of Thunder and
Laghtning,and the Effects thereof.
Z
Oxon. Fuly 26.1607.
STR july 97
] Thank you for the TranfaGtions of Fuze
_ which you fent me; wherein I am well
pleafed with Mr. Haley’s Remarks on the
Torricellian Experiment at the top of Szom-
don-hill in Wales, at the height of 1240 yards
perpendicular. Where the height of that
Quickfilver in the Barofcope was 3 Inches
and f. lefs than below at the Sea-fide; \ which
is an Obfervation of good ufe ; and would
have been more fo, had he had the leifure to
make like Obfervations at feveral other per-
-pendicular. heights in the Afcent. For
from fuch comparative Obfervations we are
to make an Eftimate, at what proportion the
height of the Quickfilver doth decreafe in re- |
ference to the height of the place. I mean
_ whether in the fame Proportion, or the Du-
plicate, Sub-duplicate, or how otherwife
Complicate thereof. From whence we may
make a Judgment of the height of the Atmof~ —
phere, if at leaft it have a determinate
height. I did once attempt (a great while
fince) a.Computation of it; but wanted a
fafficient number of Dara to proceed upon
Hh ey : ut
~
216 Mifcellanea Curiofa.
But that which is moft furprizing in thofe
TranfaGtions is, the prodigious Aas there
mentioned , which happen’d at many Places,
on different Days, and all within the compafs
of lefs than fix Weeks. I have been told of
the like in other Places about the fame time,
in Lincolnfbire, Hampfhire, and elfewhere ;
whether or no on the fame Days which you
mention, I cannot tell; nor can I give a par-
eicular Account of them. But it would be
kind in thofe who can, to give you like Ac-
counts thereof with thofe you have Publifhed,
for a like publick Information.
I find it is thought very ftrange, what
fhould caufe fo fudden a Congelation of Hail--
ftones to fo great a bignefs before they fell.
And it is indeed very ftrange. But it is not
neceflary that the whole bignefs be attained
before they begin to fall, but the freezing
may continue during the Fall, to increafe the
Bulk. For I remember that (many Years
fince) I obferved here at Oxford a ftrange —
fhower of Hail, wherein (befides the formed -
Stones that fell on the Ground, there did
hang on the Treesa great deal in the Form
of Icicles (a Foot or more in length) fo
many and heavy, as to break off fome Boughs
with their weight ; and I was then told, that
in fome places great Branches of Trees were
fo broken off; which muft needs be from the
continuing to freeze during the fall.
And truly the Generation of Hail in ge-
neral, isa thing which deferves to be farther
inquired into, than (I think) hath been yet
done. I find Mr. Haley (in his Narration)
Ae afcri-
Mifcellanea Curiofa. 317
afcribing it to Vapour difpofing the Aqueous
Parts fo to congeal.. And not unlikely.
If I may interpofe my Opinion, you may
take it thus: | |
Thunder and Lightning are fo very like the
Effects of fired Gun-powder , that we may
reafonably judge them to proceed from like ~
Caufes. The violent Explofion of Gun-
powder, attended with the Noife and Flath,
is fo like that of Thunder and Lightning, as
if they differed only as Natural and Artifi-
cial, as if Thunder and Lightning were a
_ kind of natural Gun-powder, and thisa kind
of artificial Thunder and Lightning. :
_ Now the principal Ingredients in Gun-
powder are, Nitre and Sulphur (the Admi-
ftion of Charcole being chiefly to keep the
Parts feparate for the better kindling of it.)
So that if we fuppofe in the Air, a conve-
nient mixture of Nitrous and Sulphorous Va-
pours, and thofe by Accident to take Fire ;
fuch. Explofion may well follow, with fuch
Noife and Light, asin the firing of Gun-pow-
der. And being once kindled, it will run on
from Place to Place as the Vapour leads it,
‘as in a Train of Gun-powder, with like _
Effects.
This Explofion, if high inthe Air, and far
from us, willdo no Mifchief, or not confider-
‘able; like a parcel of Gun-powder fired in ~
the open Air, where is nothing near to be
hurt by it: Butif near, to us (or among us)
it may kil] Men or Cattle, tear Trees, fire
Gunpowder, break Honfes, or the like; as
Gun-powder would do in like Circum-
ftances. 3
Now
218 Maifcellanea Curiofa.
Now this nearnefs or farnefs may be eftimated
by the Diftance of Time between feeing the
Flafh of Lightning, and hearing the Noife of
the Thunder. For though in their Genera-
tion, they be fimultaneous; yet (Light mov-
ing fafter than Sound) they come to us fuc-
ceflively. I have obferv’d that; commonly;
the Noife is about Seven or Hight Seconds
after the Flath (that is, about half a quarter
ofa Minute); but fometimes mueh fooner, in
a Second or Two, or lefs than fo; and almoft
immediately upon the Flafh. And at fuch
time, the Explofion muft needs be very near
us, or even amongft us. And, in fuch Cafes,
1 have. (more than once) prefaged the Ex-
pectation of Mifchief, and it hath proved ac-
cordingly, in the Deftru@ion of Men or Cat-
tel, andthe like. (As once at Oxford; when,
within half an Hour after fuch Prefage, I
heard of one killed at, AZedly, hard by, and
others endangered ; and another time at Tow-
cefler, when within a few Hours after, we
heard of Five Perfons kil?d at Everton, about
Four or Five Miles from us, and others
wounded ; befide other Hurt done.)
Now, thatthere is in Lightning a Sulphor-
ous Vapour, is manifeft from the Sulphorous
Smell which attends it, efpecially when Hurt
is done; and even where no Hurt is done,
fromthe Lightning it felf,more orlefs difcern- |
able. Anda fultry Heat inthe Air, is com-
monly a Fore-runner of Lightning foon after.
And that there is alfoa Nitrous Vapour
with it, we may reafonably judge, becaufe we
do not know of any Body fo liable toa fud-
dain and violent Explofion. |
Now
Mifcellanea Curiofa. 219
‘Now thefe Materials being admitted, it
remains to be confidered, how they may be
kindled in order to fuch Explofion. As to
which,I have been told from Chymitts (though
I have not feen it tried) That a Mixture of
Sulphur, Filings of Steel, with the Admiffion
of a little Water, will not only caufe a great
Effervefcence, but will of it felf break forth
into an actual Fire.
So that there wants only fome Chalybeat
_ or Vitriolick Vapour (or fomewhat equivalent)
to produce the whole Effe& (there being no
want of Aqueous Matter in the Clouds.) |
And there is no doubt, but that amongft
the various Effluvia from the Earth, there
may be copious Supplies of Matter for fach
Mixtions.
And ’tis known, that Hay, if laid up too
Green, will not only heat, but take Fire of
it felf. |
And while we are difcourfing of this, it may
fuggeft fomewhat as to the Generation of Hail
_ which is very oft an attendant of Thunder
‘and- Lightning. .’Tis well known, in our ar-
tificial Congelations, that a Mixture of Snow
and Nitre (or even common Salt) will caufe
a prefent and very fuddain Congelation of
Water. And the fame in Clouds may caufe
that of Hail-ftones. Andthe rather, beeaufe
(not only in thofe prodigioufly great, but in
common Hail-ftones) there feems fomewhat
like Snow rather than Ice, in the midft of
them. |
And, as to thofe in Particular (of which we
are now fpeaking) fo véry large (as to weigh
Halt a Pound, or Three Quarters of a Pound)
fup-
Ms a a Reid nk Se dN DN a i i ta
220 MifceHanen Curiofa. —
_ fuppofing them to fall from fo great a Height, —
as tis manifeft they did by tne Violence of
their Fall: ’Tis very poflible, that though ~
their firft Concretion, upon their fuddain
Congelation, might be but moderately great,
asin other Hail; yet, in their long Defcent,
if the AZedium through which they fall were
alike inclined to Congelation, they might —
receive a great Acceffion to their Bulk, and
divers of them incorporate into one Like as
in thofe Icicles before mentioned.
Thefe have been my Thoughts, occafioned
by the Confideration of the furprizing Great-
nefs of thefe Hail-ftones, with the great
Thunder and Lightning which did atten
thefe Storms. - | |
Yours, &c.
THE
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SYNOPSIS
Aftronomy of Comets.
HE ancient Egyptians and Chaldeans
Cif we may credit Diodorus Siculus)
by a long Courfe of Obfervations,
were able to predict the Apparitions
6f Comets. But fince they are alfo faid, by
the Help of the fame Arts, to have prognolti- ©
cated Earthquakes and Tempetts, ’tis paft ail
Doubt, that their Knowledge in thefe Matters,
was the Refult rather of meer -4ffrological Cal-
culation, than of any Aftronomical Theories of the
-Ceeleftial Motions. And the Greeks, who were
the Conquerors of both thofe People, fcarce —
found any other fort of Learning amongit
them, than this. So that ’tis to the Greeks
fhemfelves as the Inventors (and efpecially to
the Great Hipparchus) that we owe this Affro-
nomy, Which is now improv’d to fuch a Heigth.
But yet, among{t thefe, the Opinion of Ari-
ftotle (who wou’d have Comets to be nothing
elfe, but Subluaary Vapours, or Airy Meteors)
: Bes pre-
6 Mifcellanea Curiofa..
-prevail’d fo far, that this moft difficult Part
of the Aftronomical Science lay altogether neg-
lected ; for no Body thought it worth while
to take Notice of, or write about, the Wan-
dring uncertain Motions of what they efteemed
Vapours floating in the e€ther , whence it came
to pafs, that nothing certain, concerning the
Motion of Comets, can be found tranfmitted
fromthem tous. pe het af
But Seweca the Philofopker, having confider’d
the Phazoimena of Two remarkable Comets of
his Time, made no Scruple to place them a-
mongit the Celefial Bodies; believing. them
to be Stars of equal Duration with the World,
tho’ he owns their Motions to be governd by
Laws not as then known or found out. And at
laft (which was no untrue or vain Prediction)
he foretells, that there fhould be Ages fometime
hereafter, to whom Time and Diligence fhow’d
unfold all thefe Myfteries, and who fhou’d
wonder that the Ancients cou’d be ignorant of
them, aiter fome lucky Interpreter of Nature
had fhewn, zz what Parts of the Heavens the Co-
mets wanderrd, what, and how great they were.
Yet almoft all the Aftronomers differ’ from
this Opinion of Sezeca;. neither did Sezeca him-
felf think fit to fet down thofe Phenomena ot
the Motion, by which he was enabled to maintain ~
his Opinion: Nor the Times of thofe Appear-
ances, which might be of ufe to Pofterity, in
order to the Determining thefe Things. And
indeed, upon the Turning over very many
hiftorics of Comets, I find nothing at all that
can be of Service in this Affair, before, 4. D.
1337. at which time Nicephorus Gregoras, a Con-
feantinopolitar Hifterian and Aftronomer, did
oe pretty
My{cellanea Curiofa. 2
pretty accurately defcribe the Path of a Comet
amongit the Fix’d Stars, but was too laxe as
to the Account of the Time; fo that this moft
. doubtful and uncertain Comet, only deferves to
be inferted in our Catalogue, for the fake of its:
appearing near 400 Years.ago. ~
Then the next of our Comets was inthe Year
1472,Which being the {wifteft of al],and neareft
to the Earth, was obferv’d by Regiomantanus.
This Comet (fo frightful upon the Account
-bothof the Magnitude of its Body,and the Tail)
mov d Forty Degrees of a great Circle in the
Heavens, in the Space of one Day; and was
the firft, of which any proper Obfervations
are come down to us. But all thofe that
confider'd Comets, until the Time of Ticko
Brabe (that great Relftorer of Aftronomy)
believ’d them to be below the Moon, and fo
took but little Notice of them, reckoning
them no other than Vapours.
But in the Year 1577, (Ticho feriorfy
purfuing the Study of the Stars, and having
gotten Jarge Inftruments for the Performing
Cocleftial Menfurations, with far greater Care
and Certainty, than the Ancients cou’d ever
hope for) there appear’d a very remarkable
Comet ; to the Obfervation of which, Trecho
-vigoroufly applied bimfelf; and found by ma-
ny juft and faithful Trials, that it had not a
Diurnal Parallax that wasat ail perceptible:
And confequently was not only no Aircal Va-
pour, bat alfo much higher than the Moon ;
nay, might be plac’d amongit the Planets. for
‘any thing that appeard to the Contrary ,
the cavilling Oppofition made by fome of the
Bie oe A 2 School-
4 Mafcellanea Curiofa.
School-men in the medn time, being to no Pur-
pofe. eo ! hae
Next to Ticho, came the Sagacious Kepler.
He having the Advantage of Ticho’s Labours
and Obfervations, found out the true Phyfical —
Syftem of the World, and vaftly improv’d the
Aftronomical Science.. aa ae
For he demonftrated that all the Planets per-
form their Revolutions in Eliptick Orbits, whofe
Plains pafs thro’ the Center of the Sun, obferving
-‘thisLaw, That the Area’s (of rhe Elliptick Settors,
taken atthe Center of tke Sun, which he proved
to be in the common Focus of thefe Ellipfes) are
always proportional to the Times, in which the cor=
refpondent Elliptical Arches are deferib'd. He
difcover’d alfo, That the Diftances of the Pla-
nets from the Sun are in the Sefquialtera Ratio
of the “Periodical Times, or (which is all
one) That the Cubes of the Diftances are asthe —
‘Squares of the Times. This great Aftronomer —
had the Opportunity of obferving Two Co-
mets, one of which was very remarkable one. -
And from the Obfervations of thefe (which af-
forded fufficient Indications ofan Annual Paral-
lax) he concluded, That the Comets mov'd freely
‘thro? the Planetary Orbs, with a Motion vot much
different from a Rettilinear one, but of what Kind
he coud not then precifely determine. Next, He-
welius (a Noble Emulator of Ticho Brabe) fol-
lowing in Kepler’s Steps, embraced the fame
Hypothefis of the Reéctilinear Motion of Co-
mets, himfelf accurately obferving many of
them. Yet, he complaind, that his Calculations
did not perfectly agree to the Matter of Fa&
in the Heavens: And was aware, that the Path:
of a Courct was bent into a Curve Line towards the
Sarr.
| Mifcelanea Curtofa. 5
Sun. Atlength, came that prodigious Comet
of the Year 1680. which defcending (as it were)
_ from an infinite Diftance Perpendicularly towards
the Sun, arofe from him again with as great
a Velocity. ts
This Comet, (which wasfeen for Four Months
continually) by the very remarkable and pe-
culiar Curvity of its Orbit (above all o-
thers) gave the fitteft Occafion for inveftiga-
ting the Theory of the Afoticn. And the Royal
Obfervatories at Paris and Greenwich having been
for fome time founded, and committed to the
Care of molt excellent Aftronomers, the. appa-
vent Motion of this Comet was molt accurately
(perhaps as far as Humane Skill cou’d go) ob-
ferv’d by Mrs. Caffiné and Flam/teed. z
Not long after, that Great Geometrician, the —
Tlluftrious Newton, writing his Mathematical
Principles of Natural Philofophy, demonitrated
-not only that what Kepler had found, did ne-
ceflarily obtain in the Planetary Syfiem, but al-
fo, that all the Phexomena of Comets woud
naturally follow from the fame Principles;
_ which he abundantly illuftrated by the Exam-
ple of the aforefaid Comet of the Year 1680.
fhewing, at the fame time, a Method of Deli-
neating the Orbits of Comets Geometrically ;
wherein he (not without the higheft Admirati-
on of all Men) folv’d a Problem, whofe Intrica-
cy render’d it worthy of himfelf. This Comet
he prov'd to move round the Sun ina Parabo-
lical Orb, and to defcribe Area’s (taken at
the Center of the Sun) proportional to the
Times. | |
A3 Wheres
6. Mafeellanen Curtofad =
Wherefore Cfollowing the Steps of fo Great «
Man) have attempted to bring the fame Me-
thod to Arithmetical Calculation, and that with —
defired Succefs. For, having collected all the -
Obfervations of Comets I could, I fram’d
this Table, the Refult of a prodigious deal of
Calculation, which, tho’ but fmallin Bulk, will —
be no unacceptable Prefent to Aftronomers. For
‘thefe Numbers are capable of Reprefenting all
that has been yet obferv’d about the Motion of
Comets, by the Help only of the following
General Table, in the making of which I {pard
no Labour, that it might come forth perfect,
as a Thing confecrated to Pofterity, and to
jaft as long as Ajtronomy it fell. PSL
SS
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3 = _Maifcellanea Curiofa. é
A General Table for Calculating the 3
Motions of Comets in a Parabolic at
Orbit.
Med. Ang. a, Logar. Logar.
mot. |veribelio.| pro dift. pro dift.
—— |— a Sole. 4 Sole.
Oo gr, Let es # at bate
meee | eee ee ore
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3. 3.15]0,:000309-|| 32 144. 3:2010.065838
4.34 4310 000694
6. 6. O|0.001231
1£2, 7.54}0.004876
9 |i3.37.17|0.006151
10 dis, 6: 7|0:007564
ed
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12 |18. 1.54]0.010798
13 |19.28.4.7|0.012609 §§ +24 2110.10§ 752.
14 |20.54.54]0.014550 56.20.12|0.109490
15 |22.20.14 9.016607 $7:15§+ 6lo.113240}
16 |23.44.44|0.018783
17°125. 8.22]0.021072 ||
13 126.31. $19.023470 |!
19 |27.52.5«|0.025969 69.45.25]0.128278 |
20. |29.13.4710.028570|| $0 [61.35.45 10.132035
21. 130.33-4C]o.031263 (\62.25.14]0.135792
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43°133.10.231o 036916
24 134.27. 12|9.039864
_25°135-42.55]9.042892
26 136.57 411.045989
27 128.1 1.2019.049154
28 139.23.54|>.05 2382
29 |40.35-2 2.055668 |
20 141 445.4719 o8Qo09
63.13+52|0.139544
64. 1-40}o-143291
64.48.3810.147029 |
65-34-50|0.1 $0762
56 66.20 13]0,15.4482
| 57 \57 04¢50]0.1 58192 |
$5
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58 67.48-4210.16 1890]
59 168-31-50|0.165578
60 69.14.16}0-169254,
Mifcellanea Curiofa.
aun
0.172914
70.36.56]0.176557
71617.1615,180188
67 173+51-5910.194540
74.29 5]o.198085
69 175:05-38]0.201614
73 17725-51021 5 529
74 177+59-41)0.218963
75_|78+32-54 O 222378
31.41-31]lo.242416
82.11:T9!0,24.5634
82.40. 40,0.248933
83, 9.34.0.252159
85 {83.38 40-255366
84. 6. 80.2585 52
84:33:49 0.261720
83 485. 1. § 0.264865
: 85.27.58 0.267989
85.94.27 0.271092
Med.
88.01.27
Ang. | Logar.
peribelio, | pro dift
rome a Sole,
rg ae
86.20.34.0.274176
86.46.200.277239
87.11.43'0,280284
87.36.45|0,283306
0.286308
—————
83.25. 4915,289293
8.49.48'0.292252.
89.13.320.295201
89.36.54'0.298122
79.00-000.301030
90 4514/0. 306782
91.29.18)0.312469
93:34-52/0 329042
94-14-4010. 334424
94-5 3+30|0-3 39736
95-31.2210.344979
96. 8 22) 350153
| 99-44.30'0.355262
97.19.48 0.360306
97-54°1710.365284
98. 28 0010. 370200
99-99: §7,0°375052
130 | 99,33-1 110.3 79842
132 [100+ 4.430.384576
134
TOO 35-450 389252
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B 2 The
ra Maifcellanea Curiofa. ©
The Conftruction and Vfe of the
general Table.
‘As the Planets move in Elliptick Orbs, fo
do the Comets in Parabolick ones, having the
Sun in their common Focus, and defcribe equal
Area's in equal Times. But now becaufe all
Parabola’s ave fimilar to one another, therefore
if any determinate Part of the Area of a gi-
ven Parabola, be divided into any Number of
Parts at Liberty, there willbe a lke Divifion —
made in all Parabola’s, under the fame Angles,
and the Diftances will be proportional : And
confequently this one Table of ours will ferve —
for all Comets. Now, the Manner of the
Calculation of this Table is thus: In the Fig.
Let S be the Suz, POC the Orbit of a Comet,
P the Perihelion, O the Place where the Comet
is 90 gr. diftant from the Sun, C any other
Place. Draw the Right Lines CP, CS, and
makeST, SR, equal to CS; and then having ©
drawn the Right Lines CR, CT, (whereof
the one isa Tangent, and the other a Perpen-
dicular to the Curve) let fall CQ perpendicu-
lar to the 4x PSR. ie es aa
Te es easy Now,
Mifcellanea Curiofa: Fr)
_ Now, any Area, as COPS, being given, *tis
requir’d to find the Angle CSP, and the Di-
ftance CS. From the Nature of the Para-
bola RQ is ever = 2 the Parameter of the Axisy
and confequently if the Parameter be put = 2,
then RQ = 1. Let CQ=z; then PQ fhall =
z zz, and the Parabolick Segment COP=122%-
But the Triangle CSP will=4<, and fo the
_Mixtilineal Area COPS=# 2’|-4 z=, whence
zi--3 2-124. Wherefore refolving this Cu-
hical Equation, z or the Ordinate CQ. will be
known. Now, let the Area OPS be propos’d
to be divided into 100 Parts; this Area iss. of
the Square of the Parameter, and confequent-
ly 12-ais that Square—4. If therefore the
Roots of thefe Equations z*+3 z = 0, 04:0,08:
6,12: 0, 16, &c. be fucceflively extracted,
there will be obtain’d fo many z or Ordinates
CQ refpectively, and the Area SOP will be di-
vided into too Parts. And in like manner is
the Calculus to be continued beyond the Place
©. Now the Root of this Equation (fince RO
is=1) is the Tabular Tangent of the Angle
CRQ, or! the Angle CSP, and fo the Angle
CSP is given. And RC, the Secant of the fame
Angle CRQ, is a mean Proportional between
RQ or Unity, and RT, which isthe Double of |
§G, as is plain from the Conicks, But if SP be put
1 and fo the Latws Reftum — 4 (asin our Ta-
ble) then RT will be the Diftance fought, vz.
the Double of SC in the former Parabola. Af-
- ter this manner therefore, I compos’d the fore-
going Table, which ferves to reprefent the
Motions of all Comets: For hitherto there has
been none obferv’d, but comes within the Laws
of the Parabela. ,
Ie
14 WhifceHanea Curiofa,
It remains now, that we give the Rules for
the Calculation, and fhew the Way of deter-.
mining the Place of a Comet feen, by thefe
Numbers. The Velocity of a Comet moving in a
Parabola, is every where -to the Velocity of a Pla-
net defcribing aCircle about the Sun, at the fame
Diftance from the Sun, as \/ 2 to I. as appears
from Cor. 7. Prop. 16. Lib. 1. of the Princip: —
Phil. Nat. Math. Vf therefore a Comet in its
Perihelium wére fupposd to beas far diftant
from the Sun as the Earth is, then the Diurnal
Area which the Comet wou’d defcribe, wou’d
be to the Diurnal 4rea of the Earth, as {/2 to
1. And confequently, the Time of the Annual
Revolution, is to the Time in which fuch
a Comet wou’d defcribe a Quadrant of its
Orbit from the Perihelium, as 3.14159, Ces
(that is the Area of the Circle) to 4/8. There=
fore the Comet won'd defcribe that Quadrant
in 109 Days, 14 Hours, 46 Minutes; and fo
that Parabolick Area (Analogous to the Area
POS) being divided into 100 Parts, to each
Day there wou'd be alotted 0.912280. of thofe
Parts; the Log. of which, wz. 9.960128, is
to be kept for continual Ufe. Byt then the
Times in which a Comet, at a greater or lefs Di-
fiance, wowd defcribe fimilar Quadrants, are as
the Times of the Revolutions in Circles, that is, in
the Se/quiplicate Ratio of the Diftances: And
fo the Diurnal Area’s, eltimated in Centefimal
Parts of the Quadrant (which Parts we put for
Meafures of the mean Motion, like Degrees)
are in each, in the Sub/e/quialtera Proportion
of the Diftance from the Suain the Perihelion.
Thefe
— MifceHanea Curva. 1g
Thefe neceffary Things premis’d, let it be *
proposd to compute the apparent Place of aay
one of the mention’d Comets, for any Gives |
Time. Therefore, | |
1. Let the Sun’s Place be had, and the Log, of
ats Diftance from the Earth. :
2. Let the Difference between the Time of the Pe-
rihelion, aad the Time given, be gotten, in Days and
Decimal Parts of Days. Tothe Log. of this Num-
ber, let there be added the conftant Log. 9.960128,
and the Complement Arithmetical of the 2 of the
Log. of the Diftance in the Perihelium from the Sun
The Sum will be the Log. of the Mean Motion, to be
fought in the firft Column of the General Table.
3. With the Mean Motion let there be taken the
correfpendent Angle from the Perihelium, zz the
Table, aad the Log. for the Diftance from the
Sun: Then in Comets that are Dirett, add, and
in Retrograde ones fubjtratt , if the Time be after
the Perihelium, the Angle thus found, to or from
the Piace of the Perihelion ; or in Direct Comets,
fubfirat , and in Retrograde ones, add; if the
Time be before the Perihelion, the forefaid Angle
to or from the Place of the Perihelion; and fe
we {ball have the Place of the Comet in its Orbit.
. And to the Log. found for the Diftance, let there
be added the Log. of the Diftance in the Perihe-
lion, aud: the Sum will be the Log. of the true
Diftance of the Comet from the Suu.
4. The Place of the Node, together with the Place
of the Comet in its Orbit, being given, let the Di-
flance of the Comet fromthe Node be found , then,
the Inclination of the Plane being given, there will be
given alfo (fromthe common Rules of Trigonometry)
the Comet’s Place reduced to the Ecliptick, the Incli-
nation or Hleliocentrick Latitude, andthe Loz. of the
curtate Diftance. Ce
5. From
16 Mifcelanea Curiofa. ik
5. From thefe Things given (by the wei fame
Rules that we find the Planets Places, from be ag!
Place and Diftance given) we may obtain the Ap-
parent or Geocentrick Place of the Comet, toge-
ther with the Apparent Latitude. And this -it
may be worth while to illuftrate by an Exame
ple or two. sea |
ExameveE I.
Let it be requir'd to find the Place of the
Comet of the Year 1665, March 1°, 7",
oo', P. M. London. That is. 96%, 19°,
Novemb. 2h, Ar, 52".
Log. Dift. Perihels oc. o7 1044
Log. Se(quialt. o 916566
: : 9: 960128
Log. ‘Temp. te 985862
f Log. Med. Mot. 1.929424
Medius Motus 85.001
Perihel. SU 10. 41. 45
Ang. Correlp. 83. 38. os—
Comet. inOrb. GB 37+ 3° 20
' Afcend. Nod. TL 2%. 14: 00
Com. a Nodo (34. 1O- 40
Red. ad Eclip. 22 19+ OF
Com. Helioc. © i ee
Incl. Bare o37 ize 46. 50
Log. pro dif. O. 255369
Log. Perihel. QO, O11044
Co-fin. Incl. 9g 990754
Log, dift. Curt. 257197
Lor. di O >. 997918
© xX
21+ 44. 4§
Com. Vifus VW a9. 18. 3°
Lat. Vila 8. 36 15
' EXAMe
5 i eal
J = Ge ae =
81, after the Perihelion, which happen’
La ie be requir'a to
met of the Year 1683,
P.M. London: Or, 1
That 15, 214, 108,
Mifeclanee Curiofa
Examecre Il,
find the Place of i Co 0-
helion,
fini.
the
fo chavit Wa foutid if be"9
Lae
foc mo deomy Nuts, 25
Log, diff. Perihel, 9.
748343
Log. SeGuialt. 9. 622514
Comp. Arith, ©. 377486
| é 9-.950128
Log. Temp. I+ 310723
Log. Med. Mot. cass} 648337
Medius Motus 44. 498
:: ee aa | 25. 29: 30
Ang. Correfp. 56. 47. 20—
Comet. in Orb. Fs 28. des ns
Nod. Defcend. Fe ee 23. 00
Com, a Nodo 35° 19. 10
Red. ad Eclip, - 4.48. 30
Com; Helio. %€ 28.11. 30
Ancl. Bor, 35+. 2» a0
Log. pro dift; O- 111336
1 Log, Perihel. 9: 748343
“Ce: fins Incl, gies AE old
AED Hog difk Curt, g. 772866"
HL SOs) dif) OX 0958101606 194541
Wi 13:5 Locus Sy PDT A Ags 25 Oy
te eArlitle So....,
Lat. Vor. | 28 2 90
xam) ple, “twas obferwd. (at. London)
tiapp lied to the Secand St taro
ae Noi
17
July PME Sued Ue
3", 40! Equat. Time.
50’ after the Peri-
“At the, Tnftant re Tine: ‘Specified’. in pin
) that.
ries->
hetly,
and
*
i8 Mifcellanea Curiofa. |
and 3! to the Eaft, according to Mr. Hook’s Ob-
fervation. But at that of the Second Example,
Imy felf (near London, with the fame Inftru-
ments whereby I formerly obferv’d the. Sou-)
thern Conttellations) found the Place of the
Comet.to be %, 5°, 11/', and 28°, 52! North
Latitude, which agreed exaétly with the Ob-
fervation made at Greenwich almolt thé ve1y
fame Moment. | | bOT La
As for the Comet of the Year 1680, which
came almoft to the very Saw it felf (being in
its Perihelion, not.above } of the Semi-Diameter —
of the Sun diftant from the Surface of it) fince
the Latus Reftumisfo very fmall, could hardly
be contained within the Limits. of the General
Table, becanfe. of. the exceflive Velocity of
the Avean Motion. Therefore in this Comet,
the beft Way is Cafter the Atean Motion is -
found) to get from thence (by the Help of the
foregoing Equation <’- 32 =-* of the Mean
Motion) the Tangent of Half the Angle from
the Perihelion, together withthe Log. for the
Diftance from the Sun. ‘Which Things being
given, we are to proceed by the fame Rules,
as in the reft. vk |
After this Manner therefore, the Aftrono-
mical Reader tay examine. thefe Numbers,
which I have calculated, with all imaginable
Care, from the Obfervations 1 have met with.
And I have not thought fit to make them pub-
lick before they have been duly examin’d, and
made as accurate as "twas poflible, by the Study
of many, Years. 1 have publifh’d this Speci-
men of Cometical Aftronomy, as a Prodromus
of a defigned future Work, left, “happening
: (Oy Lo tae ait ye Wis CRY 2. £9
*
Mr|celanea Curiofa. 19
to die, thefe Papers might be loft, which
every Man is not capable to retrieve, by rea-
fon of the great Difficulty of the Calculation.
_Now, it may not be amifs to put the Reader
in mind, That our Five firft Comets, (the
‘Third and Fourth obferv’d by Peter Apian, the
Fifth by Paulus Fabricius) as alfo the Tenth feen
by A4e/tiin, if 1 miftake not, in the Year 15960.
are not fo certain as the reft;. for the Obferva-
tions were made neither with fit Inftruments,
nor due Care, and uponthat Account are dif-
agreeing with themfelves, and can by no means
be reconcil’d with a regular Computation. The
Comet which appear’d in the Year 1684. was
only taken Notice of by Blanchinus, who obfer-
ved at Rome: And the laft, which appear’d in
the Year 1698. was feen only by thofe at Pa-
viz, who deterimin’d its Courfe in a very un-
common Way. This Comet was very obfcure;
and, altho’ it movd fwift, and came near e-
nough our Earth; yet we, who are wont to
be curious enough in thefe Matters, faw no-
thing of it. For want of Obfervations I have
left out of the foregoing Catalogue, thofe Two
remarkable Comets which have appear’d in this
our Age, one in November, in the Year 1689 the
other in February in the Year 1702. For they di-
recting their Courfestowards the Southern Parts
of the World, and being {carce conf{picuons here
in Europe, met withno Obfervers capable of the
Bufinefs. But, ifany one fhall bring from Jrdia,
or the Southern Parts, an accurate Series of re-
quifite Obfervations, | will willingly fall to work
again; and undergo the Fatigue of reprefent-
ing their Orbits in Numbers, as 1 have done
eee )
crear. C 2 | By
20 Mifcelanea Curtofa. =
By comparing together the Accounts of the
Motions of thefe Comets, “tis apparent, their
Orbits are difpos’d in no manner of Order 5
nor can they, as the Planets are, be ne
hended, within a Zodiack, but move. indiffe-
rently every Way, as well Retrograde as Di-
rect ; from whence it is clear, they are not
carry’d about or mov’d in Vortices.. Moreover,
the Diftances in ‘their Perihelium’s are fome-
times greater, fometimes lefs ; which makes
me fufpect, there. may be a far greater Num-
ber of them, which moving in Regions more re-
mote from the Sun, become very obfcure;, and
wanting Tails, pafs by us unfeen: y caus
Hitherto I have confider’d the Orbits of
Comets as exactly Parabolick; upon which
Suppofition it wou'd follow, that Comets be-
ing impelld towards the Sun by a Centripetal
Force, defcend as from Spaces infinitely di-
ftant, and by their Falls acquire fuch a Velocity,
as that they miay again run of into the remo-
teft Parts of the Univerfe, moving upwards
with fuch a perpetual Tendency, as never to
return again to the Sun. But: fince they ap-
pear frequently enough, and fince none of them
can be found to move with an Hyperbolick
Motion, or a Motion fwifter than what the
a Comet might acquire by its Gravity to the
San, ’tis highly probable they rather move in
very Excentrick Orbits, and make their Re-
turns after long Periods of Time: For fo their
Number will be determinate, aad,. perhaps,
not fo very great. . Befides, the Space between
the Sun and the fix’d Stars is fo immenfe, that
there is Room.enough for a Comet to revolve,
tho’ the Period of its Revolution be vaftly.long.
se NOW»
MifceHanea Curtofa. 21
Now, the Latw Reflum of an Ellipfis, is to the
_ Latus Retium of a Parabola, which has the fame
- Diftance in its Perihelium; as the Diftance in
the Aphelium in the Elipfis, is to the whole
Axis of the Elipfis. And the Velocities are
in a Subduplicate Ratio of the fame: Where-
fore in very Excentrick Orbits this Ratio comes
very near to a Ratio of Equality ; and the very
{mall Difference which happens on Account of
the greater Velocity in the Parabela, is eafily
_compenfated in determining the Situation of
the Orbit. The principal Ufe therefore of
this Table of the Elements of their Motions,
and that which induced me to conftruct it, is,
That whenever a new Comet fhall appear, we
may be able to know, by comparing together
the Elements, whether it be any of thofe which
has appeard before, and confequently to deter-
- mine its Perjod, and the Axis of its Orbit,
and to foretell its Return. And, indeed, there
are many Vhings which make me believe
that the Comet which pian obferv’d in the
_ Year t93r. was the fame with that which
Kepler and Logomontanus took Notice of and
defcrib’d in the Year 1697. and which I my felf
have feen return, and obferv’d in the Year 1682.
All the Elements agree, and nothing feems ta
contradic thismy Opinion, befides the Inequali-
ty of the Periodick Revolutions: Which Inequa-
lity is not fo great neither, as that it may not be
- OWing to Phytical Caufes. For the Motion of Sz-
surmis fo difturbed by the reft of the Planets, e-
{pecially Fupiter, that the Periodick Time of that
Planet is uncertain for fome whole Days to-
gether. How much more therefore will a Co-
met be fubjeé to fuch like Errors, which rifes
Seuoee ee ale
a2 Mifcellanea Curiofa.
almoft Four times higher than Szeurz,and whofe
Velocity, tho’ encreafed but a very little, would
be fufficient to change its Orbit, from an Ellip--
tical to a Parabolical one. This, moreover,
confirms mé in my Opinion of its being the ©
fame ; that in the Year 1456. inthe Summer
time, a Comet was feen pafling Retrograde
between the Earth and the Sun, much after
the fame Manner: Which, tho’ no Body made
Obfervations upon it, yet fromits Period, and
the Manner of its Tranfit, 1 cannot think dife
ferent from thofe I have juft now mention’d.
Hence I dare venture to foretell, That it will
return again in the Year 1758. And, if it
fhould then return, we fhall have no Reafon
to doubt but the reft muft return too: There-
fore Aftronomers have a large Field to exercife
themfelves in for many Ages, before they
will be able to know the Number of thefe many
and great Bodies revolving about the common
Center of the Sun; and reduce their Motions
to certain Rules. I thought, indeed, that the —
Comet which appear’d in the Year 1532. might
be the fame with that obferv’d by Heveliusin the
Year 1661. But Apian’s Obfervations, which are
the only ones we have concerning the firft of
thefe Comets, are too rude and unskilful, for
any thing of Certainty to be drawn from them,
in fo nice a Matter. I defign to treat of ail.
thefe Things ina larger Volume, and contribute
my utmoft for the Promotion of this Part of
Aftronomy, if it fhall pleafe God to continue
my Life and Health. a chtsar ued o
In the mean time, thofe that defire to know —
how to conftruct Geometrically the Orb of a
Comet, by Three accurate Obfervations given,
, may
}
WMhi|cellanea ib eae
“may find it at the End.of the Third Book® of
‘Sir [aac Newton’s Principles of Natural Phi-
lofophy, entituled De Syflemate Mundt, in the
‘Words of its renowned Inventor. Which
“have fince been more fully. explained by my very
“worthy Collegue Dr. Gregory, in his Learned
Work of Aftronomia Phy fic ica & Geometrica.
One Thing more perhaps it may not be im-
proper or unpleafant to advertife the Aftrono-
mical Reader; That fome of thefe Comets
have their Nodes fo very near the Annual ae
‘of the Earth, that if it fhall fo happen, t
the Earth be found in the Parts of her Gi
“next the Node of fuch a Comet, whilft the.Co-
met pafles by; as the apparent Motion of the
Comet will be incredibly fwift, fo its Parallax
will become very fenfible ; and the Proportion
thereof to that of the Sun will be given. Where-
fore fuch Tranfits of Comets do afford us the
very beft Means, though they feldom happen,
to determine the Diftance of the Sun and
_ Earth: Which hitherto has only been attempt-
ed by AZars in his Oppofition to the Sun; or
elfe Venus ne Perigao , whofe Parallaxes though
triple to that of the Sun, are fcarce any ways
_ to be perceived by our Inftruments , whence
we are ftillin great Uncertainty in that Affair.
This ufe of Comets was the ingenious Thought
of that excellent Geometrician Mr. Nicolas
Facio. Now the Comet of 1472, had a Pz-
rallax above Twenty times greater than the
Sun’s. Andif the Comet of 1618, had come
down, about the Middle of March, to his de-
fcending Node: Or if that of 1684, had arr‘-_
ved a little fooner at its afiending Node;
they would have been yet much nearer the
Earthy
24 Mifcelanea Curiofa.
Earth, and confequently have had more nota-
ble Parallaxes. But hitherto none has threaten’d
the Earth with a nearer Appulfe, than that
of 1680. For by Calculation I find, that No-
“ vemb. 11°, 15, 6', P. AZ, that Comet was not
above the Semi-diameter of the the Sun to
the Northwards of the Way of the Earth.
At which Time, had the Earth been there,
the Comet would have had a Parallax equal
to that of the Moon, as I take it. ‘This is
{fpoken to Aftronomers: But what might be
the Confequences of fo near an Appulfe; or
-of a Conta& ; or, laftly, of a Shock of the
Cceleftial Bodies, (which is by no means im-
poflible to come to pafs) I leave to be dif-
_cufs'd by the Studious of Phyfical Matters.
25
Geometrical Differtation
| Concerning iiss ters nants
RAINBOW:
IN.
Which (by a dire€t Method) is fhewn
' how to find the Diameter of each
_ Bow, the Proportion of the Refraction
being given: Together with the Solu-
tion of the Izverfe Problem, or how
to fend the Ratio of the Refraéction,
the Diameter of the Iris being given.
By Ep. Hatrey, F.R.S.
of LL the Writers of Natural Hiftory,
| have particularly defcribed the Rain-
tL \ bow (a Meteor fo remarkable for its
fair Colours) and given an Account of the
Caufes of it. And the Ancient A4tholocifts,
from its wonderful Form and Appearance,
me thought
96 Mifcellanea Curtofa.
thought fit to give it the Title of Thaumanti,
or the. Child of Wonder , and placing it in the
Number of the Goddeffes, attributed to it the
Office of a Meffenger between the Celeftials
and mortal Men; which Fable, perhaps, owes
its Original to Gevefis, Ch.9. V.13. i
Thofe that attentively confider’d the Phe- —
nomena of the Rainbow, always found, that
the Sun’s Rays reflected by a Watery Cloud,
came to the Eye under a certain Angle; from
whence arofe the Arch, or Circular Figure of
it. But as for the Caufe and Reafon of the
Colours, as alfo of the’ Magnitude of the
Angle, by which we conftantly find it diftant
from the Point oppofite ro the Sun; thefe were
Things, that a long while, and very greatly
perlex’d, as well the A/oderus, as Ancients.
Neither did they doany thing to the Purpofe
herein, till the Famous Monfieur Des Cartes
making ufe of the Mathematical Sciences,
fhew’d by feveral Examples, that more ftrict
and clofe Methods of Reafoning might and
ought to obtain, even in our Management of
thofe Phyfical Speculations. Amongft other
things (tho’ it muft be own’d that herein he
had fome Light, from the Learned Antonio de
Dominis, Arch-bifhop of Spalato) he explain’d — .
the Theory of the Rainbow. And having dif-
cover’d the Laws of Refraction, he clearly de-
monftrated, that the Primary Ir was nothing
elfe, but the Sun’s Image reflexed from the
Concave Surface of innumerable Spherical
Drops of Rain ; and that with this Condition,
that thofe Rays that were parallel at their
Incidence, were not loft or diflipated by the
Reflexion, and the Two Refrattions (one at the
age . | la-
e
Mifcellanea Curiofa. 27
Ingrefs, and the other at the Egrefs) but fell
(and that alfo parallel) on the Eye. That the
_Rays were tingd with Colours by thofe Re-
fractions, after the fame manner as we fee they
are bya Gla/s Prifme. That the Secundary Iris
is produced, after the fame manner, by the
Rays that fall more obliquely, only here are
Two Reflexions, before the Sun’s Rays (which
when refracted a Second time proceed parallel
to the Eye) emerge out of the Drops of Wa-
ter. Further, that the Magnitude of each Jris
depends upon the Degrees of the Refraction,
which is different according to the Nature of
each tranf{parent Solid or Liquid. ;
And fuppofing the Proportion of the Sines
of the Angles of Incidence to the Sines of the
refracted Angles, to be in Water, as 250, to
187, he determin’d the Semi-Diameter of each
Iris, agreeably to Obfervations, viz. that of
the Primary Ir, 41°. 30'. and that of the
Secundary, 51°. 54'. By which he did not fo
much confirm the Theory it felf, which was
_ demonftrated from other Principles, as the
— Truth of the fore-mention’d affumed Proportion,
(viz. that of the Refraction.) But for thefe
Things, the Reader may confult the 8th Chap-
ter of Cartes’s Meteors, whither I refer him.
But now Cartes (who ufed an indire& and
tentative Method in determining thefe Angles)
did not feem clearly to apprehend the Ealinefs
of the Problem he had propofed to himfelf.
And becaufe none (that | know of) fince him,
has handled the fame Argument more fully;
and alfo fince fome have mifunderftood what
Cartes did, committing very great Paralogifms,
in fome Books (fince his time) which ao
larly
28 | Mifcellanea Curiofa,
larly sindbis to explain the Phenomena of
the Rainbow; I was’ willing to fupply- what 1 _
thought was ” wanting in this Dodrine, and
from the Proportion of the Refrattion given, Geo-
metrically to determine the Angle of its Di-
ftance from the Point oppofite ro the Suw: Or
contrarywife, from the Irs given, to determine
the refractive Power of the Liquid.
What the Celebrated Mr. Newton has done
upon this Head, the Reader will find (with
much greater Advantage) i in his Book of Light
and Colours, when he fhall think fit to be-
ftow thofe excellent Difcoveries upon the Pub-
lick.
But to proceed. ?Tis plain from what Cartes
has demonftrated, that the Primary Iris is
form’d by fuch Rays of the Sun, where the
Excefs of Two refratted Angles, above one
Angle of Jacidence, is the Greateft of all fuck
Exceffes poffible. And that the Secondary Ives is
form’d by thofe Rays only, where the Excefs
of Three refrated Angles, above one of Inci-
dence, 1s in like manner the Greateft. And fo
we may goon toa 3d, 4th, or any other Is,
which are form’d, where the Rays emerge af- —
ter 3, 4, 0Fr more Reflexions. Bat thefe can
never be feen in the Heavens, becaufe of the
sun’s Light which is ftill more “and more debi-.
litated by each Reflexion and Refraétion ;
Whence it comes to pafs alfo, that the Secon-
dary Ive, is painted with Colours, fo. much
faiater than the Primary one.“ But in all thefe
the general Rule is, that the Excefs of 4, or 5,
or more refratted Angles, (viz. the Number af
Refiexions being: increafed by Unity) above one
vingle of Incidence, w% of all the Greatelf.
Now: —
Mifcellanea Curiofa. 29 ©
Now this greate/? Exce/s doubled, is always
the Diftance of the Jris from the Point oppofite
to the Sun, when the Number of Reflexions is
uneven. But if that Number be eve, then the
Double of that greateft Angle, is the Diftance
of the Iris from the Sun it felf, viz. in the 2d,
4th, oth, Gc. Iris. All thefe Things are ei-
ther purely Cartefizs’s, or elfe eafily follow
from his Writings in the foremention’d Place.
But now to obtain thofe greateft Exceffes,
having the Refra@ion of any Liquor given ;
tis to be obferv’d, that the Excefs of Two
refraded Angles, above one of Incidence, is
there the Greate/?, where the momentancous In-
crement of the Angle of Incidence is exa@ly
double of the momentaneous Increment of the re-
fracted Angle.. And:that the Excefs of Three
refracted Angles is there the Greate/?, where
the Increment of the Angle of Incidence is
triple the Increment of the refracted Angle; —
and fo of the reft. And this is fufficiently
evident of it felf: But asfor the Angles, we
may obtain them by the Heip of the following:
eid dade muff therefore be demonftra-
ted. .
Ewa
20 WVhifcellanea Curiofa. so
LEM™ A.
The. Legs. of any plain Triangle continuing ;
af the Vertical Angle be au mented or
diminifb’d, by an Angle lee than any
Angle affign’d , the Momenta or Inftan--
taneous Mutations of the Angles at the
Bafe, are to one another reciprocally, as
the Segments of the Bafe. |
At Fig. 1. Plate 3. fuppofe the Triangle
ABC, whofe Vertex is 4, its Legs AB, AC,
and Bafe BC, upon which let fall the Perpendi-
cular AD. .Then let the Angle BAC be in-
creafed by the Indivifible Momentum CAc, and
let the Lines Bcd, c D be drawn, which dif-
fer, in Imagination only, from the Lines BCD,
CD. fay, that the Aomentum of the Angle
ABC (viz. CBe) is to the Afomentum of the
Angle ACB or ACD, as CDto BD, that is
reciprocally as the Segments of the Bafe.
DEMONSTRATION.
Becaufe the Angle ACD, is the Sumof the
Angles ABC, BAC, its Adomentum alfo fhall
equal the Sum of the AZomenta of thofe Angles ;
that is, it fhall equal CAc-+-CBe. But CAc
= CDe, fince, becanfe of the right Angle at
D, the Points A,D,C, c, are all in the Arch
of aCircle, whofe Diameter is AC: By Eucl. .
3.9. And confequently the Sum of the Angles
CBc, CDec (that is the Angle Ded) hall be
the Momentum of the Angle ACD or ACB.
But
Mifcellanes. Curiofa.. 31
But thofe Angles CBc, Ded, being indefinite-
ly fmall, are to one another-as their oppofite
fides, that is, as eD orCDto BD, that is as
the Segments of the Bafe reciprocally. Q:
_ If each of the Angles Band C be Acute, the. .
Lemma will {till (mutate mutandis) be demon-
- rated after the fame Manner. ~~ : |
COROLLARY.
Hence it follows that the Avomenta of the
_ Angles at the Bafe, are to one another direttly,
as the T'angents of thofe Angles. |
By the Help of this Lemma, I will be eafie
to find the Diameter of any Jris whatfoever 5
and that either by Calculation, or a Geometrical
Conftruttion, For taking any tight Line, as
CA (Fig..2.) let it be divided firft of all in
D, fo that CA, may be toCD, in the Ratio
of the Refraction in Water, which is as 250
to. 187, or more accurately, as §29 to 396.
Then let CA be divided fo in E, that CE may)
be to AE, as Unity to the Number of Reflexi-.
ons, a Ray of the Sun (fit to produce. the Tris
propofed) undergoes: And upon the Diame-
ter AE defcribing the Semi-Circle ABE, on
the Center C with the Radius CD defcribe the
Arch BD, meeting the Semi-Circle ABE in
the Point B. Laitly, Drawing the Right Lines
2B, AB, let CF be let fall perpendicular upon
AB produced, and EB paralfel thereto. | fay
then, that CBF is the Angle of Incidence, and
CAB the Refracted Angle that we enquire af-
ter, and: which will produce the Js pro-
eS, aaa D E-
32 Mafcellanea Curiofa.
DEMONSTRATION. —
Becaufe the Triangles ACF, AEB are fimi-
lar, it will be AF: BF:: AC:EC; that is,
asthe Number of Reflexions-encreas'd by Uni-
ty to Unity (by the Conftruttion) and confe-
quently the A¢omentum of the Angle CBF, will
be to the Afomentum of the Angle CAF, in the
fame Proportion (by the foregoing Lemma.)
But the Sine of the Angle CBF, is to the
Sine of the Angle CAF, in the Proportion of
the Sides CA, CB, that is, in the Proportion
of the Refraction given (alfo by the Conftruttion.)
Therefore CAF is the Refracted Angle, cor-
refponding to the Angle of Incidence CBF;
and their Afomenta are in the Ratio propos’d, *
wherefore they are the Angles fought. Q.E.D. |
And now, multiplying the Refracted An-
gle by the Number of the Reflexions encreas’d
by Unity, and from the Product fubftraéing
the Angle of Incidence, we fhall have half the
Diftance of the Wis from. the’ Sun, if ‘the
Number of Reflexions be even, or from the
Point oppofite to the Sun, if that Number be
uneven, as we have fhewn already. Hence
we may exhibit (by a Confrattion concife and
eloquent enough) the Incidencies of all the
Orders of iris’s, td -any Liquor whofe Refra-*
ction isknown. For if theLLine AC (FIG. 2,)
be divided into: Two eqnal Parts at E, into:
Three equal Parts at ¢, into Four at <, inté:
Five atv, &c. And on the Diameter AE, ‘Aé,.
Az, An, be defcrib'd, the Semi-Circles ABE,”
Abe, Ape, Aun; which are~ all’ interfe@ted cin’
the Points B, b, 6, v, bythe Arch DBbéu,;-de=
Bi ah ferib’d
Pataca
Pa =p ee
es
Mifcellanea Charcofis igs
4 -ferib’d on the Center C with the Radius CDo.
which is to AC, in the given Proportion of
of the Refracted Angle V
the RefraGion: I fay then that the Lines
AB, Ab, A8, Av, will make with the Line
AG, the Angles CAB, CAb, CA3, CAv, equal
to the Refradted Angles; and with the re-
fpective Rayes CB, Cb, C3, Cv, they will
make Angles equal to the Angles of Inci-
dence that are required ; viz. ABC (or ra-
ther its complement to a Semicircle) for the
Primary Iris, AbC, for the Secondary, ABC,
for a Third Tri, AvC, for a Fourth, If any
one has a mind, to find thefe Angles by an
accurate Calculation, twill follow from the
fame Principle, that putting the Radius—t,
and the Ratio of the Refraction as; to s, the
eee eee
: Sine of Incidence will be ¥: Si OS aad the
| 3
| 355,
Sine of the Refra&ed Angle V4! fom
Beit o>
which Angles proceeds the Primary Iris. For
the Secondary the Sine of Incidence wiil be
, - S I
eae ‘and the Sine of the Refracted
Bi,
Angle joel = Fora Third Tris, the Sine
err eB;
of Incidence will be V16 oR aed. and the Siné
f , ig? wes
1655. rt
——-—-. Fora
19rr 15.
D | | Fourth ;
eS Pee eee eee Raha hy
Le
24. Mifcellanea Curiofa, -
Fourth Iris, the Sine of Incidence will be
V25_ +" and the Sine of the Refracted
24; DAMS. 63 :
aad oheag and in like manner of
247r 24 ! :
thereft. Farther, ’twill be found by Calcu-
lation, that (taking Cartes’s Proportion) the
Primary Tris is diftant 41°. 30. from the
Point oppofite to the Sun; the Secondary,
g1°.§5- fromthe fame. The Third, 40°. 20.
and the Fourth, 45°. 33- from the Sun it felf;
which Jrés’s perhaps were hardly ever feen for
the reafons before mentioned. —>_—.
And thus much may fuffice concerning the
Magnitude of the Jrides, in the perfpicuous
Drops of a Fluid, whofe Refractive Power is
known. It remains that nothing be faid con-
cerning the Colours, which this Phenomenon
prefents, with the orders of them in each
fort of Jris, according to all the poffible Vari-
ations of the Refraction. ‘
And here we muft know efpecially, that
the Acute and Sagacious Mr. Newton, has
found by moft clear Experiments, that the
Rays of Light are not Simple and Uniform,* as
they iffue out of the Luminous Body, but the pure
white Light which we fee, confifts of Corpufcles of
all kinds of Colours, mix’d and hurried with
a violent Motion, one amongft another.. And
that the diverfity of the Colours of things
arifes, according to the various Difpofitions
thofe Objects have, to Refract or Reflect this
or that peculiar kind of Light. Maoh
The Proof of which is manifeft from Refra-
: ctions,
Mifcelanea Curiofa. 35
tions, in which thefe Species are feparated
from one another, and the Blue or Purple
Light, (even in the fame Diaphanous Body)
is more Refracted than the Ye#om or Red. But
_ let the Reader confult this incomparable Per-
fon’s Letters (N°. 80. and the following of the
Philofophical Tranfattions) from which Speci-
-men he will be able to judge, how nobly this
Argument of Light will be managed by him.
To my purpofe ’tis fufficient, that all kinds
of Blue Light, are fomething more refrated
than Red, from which difference arifes the
Latitude of the Jrides, which is hardly to be
determined by Obfervation, becaufe of the
uncertain Limits of the Colours. But by how
much the Proportion between CA and CD, is
of greater Inequality, or by how much the Re=
fraction is greater, fo much the greater is the
diftance of any Jris from the Sun, and confe-
quently thofe borders that are remoter from
the Sun, fhine witha Purple Colour, but thofe
that are nearer, with an intenfe Red.
This may always be feen in the Primary
Jris, which vanifhes in the part oppofite to
the Sun, if the Sine of Incidence be to the
Sine of the Refraéted Angle, as CA to CE, _
“or as2tor. But if that Ratio be greater,
there can be no Primary Jris feen at all. i:
As for the Secondary Iris, ’tis to be noted; —
that this vanifhes into a Point, in the part
oppofite to the Sun, when the Ratio of the
pay ak : nnt
' Refraction is as 1 to yan —. vs , .or'as.t
to 0847487... and from thence it returns
back to the Sum it felf, where it vanifhes, if
the faid Ratio be as 3 to 1, or as CA to Ce:
D 2 But
rk » 3 ; ~ 4 ot ST) AEP RE Pe oe at eee Oe ee
36 Mifcellanea Curiofa.
But in the Ratio’s between thefe (fuch as we -
have in all Fluids known to us, except the
Air) by how much the greater is the Ratio,
by fo much is the Js more diftant from the
Point oppofite to the Sun, or rather from the ~
Sun it felf, reckoning the Arch beyond a
Semicircle. And confequently the Colours
will feem to be in a different order from the
| Primary Iris, in thefe returnings, unlefs the
diftance of the [vides from the Sun, be taken
in this Sence, which is alfo every where to be
obferved in the reft.
The Third Iris is confafed in the part op-
at to the Sun, the Ratio of the Refraction —
eing as 1 to,91855--- from thence it re-
turns. back to the Sun in the Ratio of 1 to
,68250--- whence again, the order of the
Colours being reftored,in the Ratio of 4 to 1,
or CA to Ce, it terminates in the part oppo-
fite to the Sun. The Fourth Jvs beginning
from the Sun, in a Ratio of Equality, pafles
on to the oppofite Point, in the Ratio of 1 to
,594895--- andthence returns back to theSun, ©
if the Proportion be as 5 to 43 hence again,
it difperfes*to the Point oppofite to the Sun
in the Ratio of 1 to ,55337---, within which
_compafs are included the Refradtions of all
Fluids that are known. Laftly, The Ratio |
-being as 5 to 1, or CA toCn, it vanifhes in
the very Sun it felf; the Colours being every .
where cnverted to the fight in its return to
the Sun, as they were ereét in its egref/s from ~
It. Hlence, in watery Clouds, the Br/?t and
Fourth Iris fhew deep red Colours-turned to-
wards the Sun; but the Second and Third give ©
Purple, But perhaps I may feem too tedious
“in
Muifcelanea Curiofa, 27
in thefe Defcriptions, the Rainbow it felf. bes
ing no more than a Momentary Phantafm...
_ But whence ’tis that the different Refra-
clive Power of Fluids arifes, is a Problem of
the greateft Moment » and to,be rank’d
amoneft the Secrets of Nature, not yet. ob-
vious either to our Sences.or our Reafonings.
For pure Water amoneft all. Fluids, . does
leaft of all Refra& the Rays of Light. When
tis Tinfur'd with Salts diffolved in it, accor-
ding to its weight and the quantity of Salt,
it increafes the Refra@ions. . And Corr ofive
Spirits (which. are much heavier than Water)
do alfo much more Refra& the- Rays. of
Light: Nor is it any wonder, fince being
denfer Bodies, they may eafily be conceived
fo much the more to obftrué the paflage of
the Rays. But why there fhould be f great
a Refraction in Burning Spirits and Oils, efpe-'
cially in Spirit of Turpentine, or of Wine, {ince
_ they are Fluids extreamly Light in comparifon
of Water, and confift very much of fubtle
fétherial Particles, does. not fo eafily appear’;
but feems,to. require (in order to the Expli-
cation of it) a more thorough knowledge of
the Nature and Texture of Light.
But. from the diftance (of the Jvs from the
Sun) giver, to find the Ratio of the Refra-
ction, is a thing that will give thofe that are
curious, an occafion of finding the RefraAion
of any Fluid, accurately and with little trou-
ble. For if a fmall drop of any tran{parent
Fluid, be fuppofed to hang at the bottom of
a {mall Glafs Tube, and the Sun being near
the Horizon and fhining ftrongly, it be ob-
ferved under what Angle (with the Poize
eHIEITD 3 oppofite ~
38 Maifcellanea Curiofa,
oppofite to the Sun) the Colours of the Jris be
feen in the drop, then the Proportion fought
_ will be obtained with a little Calculation. It
is a Cubical Equation, explicable by one only
Root, by which, from the Primary Iris given, the
Ratio is computed, wiz. T? 43 T?t—4rrt=o,
where T is the Tangent of the Angle of Jnci-
dence requifite, t the Tangent of $ the diftance
of the Iris from the Point oppofite to the Sun,
to the Radiusv=1. Whence (according to
Cardanus’s Rules) arifes this Theorem, viz.
From the Cube of t fubftratt the Produ of 2tr
anto the Excefs of the Secant of the fame Arch
above the Radius; the difference {hall be the leffer
Cube. The Sum of the fame, adding atrr, will
be the greater Cube. The Sum of the fides of both
Cubes, and of t, will be equal to the Tangent of
the Angle of Incidence, and the half of that, will.
be the Tangent of the Refratted Angle. From —
whence the Ratio. fought-is manifeft.
Kor an Example of this. In adrop of Oi of
‘Turpentine, the diftance of the Primary Iris; ©
from the Point oppofite to the Sun, is ob-
ferved to be 25°. 40’. °Tis required to find
the Ratio of the Refraétion.
©, 2278063
t=Tahey \12°i - go. hes
s=Sec. of the fame.’ = 1, 0256197. ~
ttt oe) SO OL a aay |
pert tT TB ; : Ly
SE X.2E0. = 05 OLIGIIOS V,-
The Difference is the lefler Cube Q; 0001495 2
: whofe fide 0,0530773 .
The Sum 0, 02349482, Osea Ht, ae
4trr 0, 91122525. | :
Re |
ed
Greatey
— MifceNanea Curiofa. 29
Greater Cube o, 93472007, whole fide 0, 9777486
ah : t 0, 2278063
et ne. liicid. 51°. 32’. 1, 2586322
2 Beane. Ref. 52°. 11’. 0, 6293161
_ . Laftly, AsVTT-4' VTT Oe poe
12168026. Which Proportion comes very
near to that, which Experience fhews to be
in Glafs and moft pellucid Solids. The Dia-
mond indeed, exceeds all tranfparent Bodies,
- Hot only in refpect of its hardnefs and value,
but alfo its RefraGtive Power, the Propor-
tion here being as 5:2, nearly, or more ac-
curately as 100:41. But of this, perhaps
more in another place. |
While I was writing thefe things, that
_ skillful Geometrician Mr. De Moivre, at my
requeft, found a like Equation for deter- |
mining the Ratio, from the Semidiameter of
the Secondary Iris, given. By which, the Ratio
is indeed fomething more exadly determined,
but that Equation being a Biquadratical one,
the Calculation is not fo ealily performed.
This Equation is T*-- 4 T? t—2 T? 1r?— 3
_ r*=0;3 where Tis the Tangent of the Re-
fracted Angle, t, the Tangent of 3 the di-
ftance of the Iris from the Point oppofite to
the Sun, to the Radius r=1. And this Equa-
tion is of that Form, as to be always expli-
cable, by an Affirmative and one Negative
Root, the one and the lefs of which, is the
Tangent of the Refracted Angle, in the Ke-
grefs to the Sun, viz. when the Purple Colours
are nearer to the Sun. The greater Root is
the Tangent of the Refracted Angie in an
: 4 Iris
~
4Q Niheallnas Curio! a
Iris. Lone out. from the Sun, viz ei in m
a lefs Ratio. An Oil of Turpentir ne, the Pe hl
af ‘cis | ‘Tris from the. Point oppolite the
Sun,.. 1s. obferved. to be 81°. 30~ whe ace the
curious, Reader may find ‘out the ‘Roots
&, 80822,-- and—2, 98131-- the Tangents of
the Refraéted Angles. Hencetis computed the
Ratio of greater Inequality, as 1 to 0, 67!
fuch as is in Oil of ‘Turpentine. ‘But: ‘from |
the greater Root comes forth the leffer Ratio,
as. 1.0 0,.9§40 nearly, fuch as would be in.
a Fluid, exhibiting. a Secondary. Tri : Of the
fame Diameter, but which (after the manner
of the Primary ‘one) fhould look towal rds. the
Sun with the Red Colours, ia <
‘Tf.any one has a mind to find inehs ROG
by a Geometrical Conftruétion, any Parabola
being given, it is done with fo much eafe,
that I need not repeat what I have already
offered. apDA that Head Philofophical Tranf-
uttions, N. 188.
Bach of thefe Equations is deduced from
what has been faid before, and alfo from ©
the Rules for, the Tangents of the Double
and ‘Treble’ Arch $s “the yap hint of which,
may be inftead of a Demonftration even fo
- thofe that are but meanly VaR in iit
thia oa
This Difcourfe. being already in the Fifi.
there came to my hands (by the. means of q
Friend) a certain Book; whofe .Title was.
Thaumantiadis Thaumafia , printed cat. Norim-
berg 1699, nines the Superintendency of AL.
Sturmius. Ya which the skillful Author feems
to have laid together whatever is to be found
of
2
Mifcellanea Curiofa. a1
of this Argument, as well amongft the Af-
ese as the ~dncient Writers ; fubjoining and
tCartes'’s, Eckard’s, "Honoratus Faber’ S,
and Mariott s Calculus. From whence it is plain,
that the reft added very little or nothing to
Cartes’s Inventions, building upon the fame Un-
geometrical and Tentative Methods of Calcula-
‘tion. But that the judicious Reader may be
Menfible, what things Ihave performed in the Do-
Grine of the iris, { would have him read the
fore-mentioned Book, and compare it with this
Difcourfe ; left in putting out thefe things, I
Soe feem only to have made an unpleafing
ition of what had been done before. And
© what waft Use in Aftronomical Matters, this
Lemma of ours may be, fhall be fhewn upon fome
other Occafion. —
B THE
42 Fre. da,¥ asin
Rey a Be Pe
~ ee |
Stereagrapbi 2 kb Projtion,
f fj e +6 1
‘Laid down and Demonftrated.
. By Fa. Hodg fon, F. RS.
DEFINITIONS,
IRCLES deferib’d upon the Surface of
the Sphere, are either great or fmall,
Great Circles are thofe which divide the
Sica of the Sphere into equal Parts; or
through which, if the Globe be cut by a Plain,
it’ will be divided equally.
_ 2. Small Circles are thofe which cut the Sur-
face of the Sphere unequally ;, or thro’ which,
if the Globe be cut by a Plain, it will ie a diyi-
ded i Into unequal Parts. ee
3. . The
Mifcellanea. Cuviofa. 43
_ 3 The Pole of any Circle, whether Great
or Small, is that Point upon the Surface of the
Sphere, which is every way equally diftant from
the Circumference. :
Whence it follows.
‘yCorol. 1. That-every Circlesdefcribed upon
the Superficies of the Globe has 2 real Poles.
__ And fince every fmall Circle is equi-diftant —
from fome one great Circle, it follows,
2. That the Poles of every fmall Circle are
the fame with the Poles of thofe great Circles,
to which they ate parallel. a
_. 3+ Wherefore all fmall Circles parallel to
each other, have the fame common Pole,
~ General Definition.
Conceive the Eye placed fomewhere in the
Superficiés of the Globe, and at the fame time
a-Plain-to-cut the Globe, and to ftand at right
Angles to the Line connecting the Centers of
the Globe and Eye. . If infinite right Lines be
imagin’d to flow from the Eye, to every Point
an the Circumference. of, every Circle defcrib’d
‘on its Surface, they will trace out upon the
Cutting Plain what is, called a Stereographick
Projection of the Sphere. ba «3
‘Whence, and’ from the firft Corollary of the
42,2, Definitions, it follows,
e : Corl, I That the Eye is placed in one of the
Poles of that great Circle through which the
Plain pafles, and upon which the Projedion is
44 Mifcellanes® }
to be formed which, ‘for Ditinaions Sake>
is, called the Plain of Projettion ; ‘and the. Pole
CARTES to the Eye, the remoter Pole,”
. That the Pole, when projeéted, ‘falls in
ae Center of the Plain of Projection,
3. That All great Circles interfedting each o
ther in the Eye-Point, will, when projee
“become right Lines, interfecting each: th 1
the Center or Pole of Projection, ‘inafmucli
~ they all lye in the fame Plain with the Eye.
4. That the Poles of every fuch Circle will
be found in the Circumference of the Plain‘of
_Projeétion, at 90 Degrees. diftance from ‘ ‘the
Gircles Interfeétion. |
5. That the Lines flowing frouf thie Bye to
any other Circle whatfoever, form a Conical
Superficies.
And in order to enquire into their Nature
and Properties when projected, it is: necellary
to. premife the two following. Lininia’s, ef v4
“LEMMA the It.
If an “ape Cone be cut with a Plain, pa-
_ rallel to the Bafe, the Section made in the Super-
ficies of the Cone will be the Circumference of
a Circle, whofe Center will be found'in the
right Line, connecting the Vertex of the Cone,
and Center Of ats’ Bates
Whence, and from the pelea Definition, it
follows, ap saci
Tos
Corol. 1. That all fmall Circles pavaltel eb 5 the
Plain of Projetion, will become Cireles in the
Projection.
2. That
Mifcellenea Curiofe. — 45
soi That.they, will all have one common Cen.
oter, viz. the Center or Pole of Projection. on
3. That their common Center, viz. the
Centre of Projection, will be their common
Pole.
20 dae ‘That their Radij or Semi-__ ig. Ie
‘diameters will be equal to the
Semi-tangent of their Diftance from the rema-
» tek Pole of Projection.
For in Figure 1, fuppofing ¢ the Eye, ab
(the Plain .of Projection, p its remoter Pole,
»cand mn, the Circle to be projected; oc equal
sito gq, will be equal to the Semi-tangent of mp
or pn, the Diftance of the Circle mm im trom p-the
03 remater Pole.
~ LEMMA the Id,
tl a an inclined Cone be cut with a Plain 3 not
i elas, to.the Bafe, yet fo that it cuts off a
Part fimilar to the whole, the Se¢tion made i in
the Superficies of the Cone fhall be a Circle.
Whence it follows.
do That all Circles. whatfoever ( except
aiff fach as lie in the fame Plain with the Eye, and
“eine which have been already confidered) tho they
are not parallel to the Periphery of the Plain
of Projection, yet become Circles in the Pro-
i: je@ion..
For if, as before, e reprefents Fig. 2, 344:
the Place of the Eye, ab the Plain
a » of; Projection, p its remoter Pole, and mn the
- Circle.to be projected.
wiT ss Through
46 Mifcellanea Curiofa,
ace _ Through 3 n draw nr parallel to.
Fig. 2, 3,4. “ab, thenis the Angle rae equal to.
| emn and men, common to both: z
Wherefore the Triangles’ émn and eny, and
confequently eoq, are fimilar, and oq a Circle Te by.
the 2d Lemina. | ss |
_ Wherefore, a 4: pdt 10 Dios
Fig. 2, 3, 4. 2. That of all’ the pibnglests: of
any one of thofe great Circles to”
be projected, their Centers and Poles will be.
found in that projected Diameter, which con-
nects the two InterfeCtions of the Circle to. be,
projected; and that other great Circle: which.
a through the Eye, and cuts the Circle to’
€ projected at right Angles.
DEFLN BED Bp
Which for Brevity’ s Sake i is called biel Pros
jected Axis: Thus oq is the projected Axis.
3. That all fmall Circles: will have’ their.
projected Axis’s in that Line, in which the pro-.
jected Axis of the great Circle lies, to. which,
. they are parallel.
“4 That the Semitangents of the preateft.
and leaft Diftances of any Circle from the re-.
moter ‘Pole of Projection, fet off eitlier on. “the.
fame or contrary fides of the Center of Proje&ti-..
on, as the Cafe directs, will give the Interfecti-"
ons or Extremities of the projected Axis,
~ 5. That of every great. or fmall Circle.
within which the Pole of Projection lies, hele,
Diameter or projected Axis is equal to
Sum of the Semi-tangents of their great
an
— Mifcellanea Curiofa. 49
and leaft Diftances .from, the Pole of. Proje-
» Thus in Fig. 2. and. 3. 0g equal to oc and ¢q
together, is equal tothe Sum of the Semi-tan-
gents of the Arches mp pn, the greateft and
leaft Diftance of the Circle from the remoter
Pole of ProyeCtion;.....cagatyror io é
6. That of every {mall Circle without which
the Pole of Projection lies; its Diameter or
projected Axis is equal to the Difference of its
greateft and. leaft Diftances from the Pole of
ProjeCion.
_ Thusin Fig. 4. 0q is equal to the Difference
between ¢q and co, the Semi-tangents of pn, and
pm, the greateft and leaft Diftances of the Cir-
cle fromthe Pole of Projection.
~, That of the a Poles of every great Circle,
and confequently of all fmall Circles parallel
to it, that which falls within the Plain of Pro-
jection will be diftant from the Center by. the
Semi-tangent.of the Excefs of the greateft Di-
ftance, of the projected. great Circle from the
remoter Pole of Projection above a Quadrant,
or the DeféG of the leaft Diftance from the Pole
of Projection to a Quadrant. | é
Thus in Fig.2. d the Pole of the Circle um
is diftant from the Center by the Space of dc,
the Semi-tangent of the Arches xp the Excefs
of pm above a Quadrant, or the Defeé of pr
to a Quadrant. |
_ 8. Thatits other Pole will be in the projected
Axis on the contrary fide of the Centre, and
diftant from ,it by the Semi-tangent of the pro-
jected Circle’s neareft Diftance from the remotes
Pole of Projection, Jeffen’d by a Quadrant..
9. Hence
t
48 Mifeellanea Curiofa, =
9. Hence we are taught how to find the in-
terior Pole of any great Circle, and confequent-
ly of all {mall Circles parallel to it, by fetting
of the Semi-tangent of the Compliment of its
neareft Diftance from the remoter Pole of Pro>- —
jection in its projected Axis, on the contra-
ry Side of the Center. with the Interfe-
ction. , DEES AOE
10. For the exterior Pole, by fetting of the
Semi-tangent of its neareft Diftance augmen-
ted by a Quadrant in the projegted Axis
from the Center, on the fame fide with the In-
terfection. lait Be IN hy (hi
PROPOSITION E._
If two Circles cab and bag interfec& each o-
ther in the Point a, the Angle gab formed by
them at their Interfection will be equal to the
Angle bse made by the Radii ab and ac, drawn
to the Point of Interfe@tion 4. is
' Conftr. To the Point of Interfection 4, draw —
the Line fa a Tangent to the Circle gab, and dz
a Tangent to the Circle hac. fab i
' Demonjftr. Becaufe the infinitely Fig. 5.
{mall Portions of the Circles gab |
and hac do coincede with theTangents fa and da,
and confequently have the fame Direction ;
therefore the curv’d lin’d Angle gab is equal to |
the right-lin’d Angle fad, formed by the Tan-
gents f2 and da: And becaufe the Angle fabis
equal to dac, take away from each the interja |
cent Angle dab, and there willremain the An- 7
gle bac equal to fad, equal to gab, which was
to be proved, "8 eee
PRQ-.
C2 ox
PROPOSITION I.
| The Angles made. by the Circles on the
Surface of the Sphere, are equal to thofe made
by their Reprefentatives on the Plain of Pro-
MR iees sarki - |
The Reafon of which is evident from the ge-
neral Definition and laft Propo- :
fition: For (in Fig. 6.) let e Fig, 6.
reprefent the Eye, bc the Plain
of Projection, ope the Angle to be projegted,
draw pn parallel to bc, pd, and pf, Tangents to
the Circles pe and po, and continne them till
they meet the Plain be, in the Line df ;
Becaufe the. Angle dpm is equal to npm, and
ie equal to mpm, therefore is md equal to
Wherefore in the Triangles dmp and fmd are
are to dm, df common, and the Angles
fam, fdp both right ; therefore the Angle dof is
equal to the Angle fmd. Therefore, &'c,
Whence, and from the firft Propofition,
it follows,
- 4. That if every great Cir- Fig. 7.
cle to be projected (which does |
not lie in the Plain of the Eye) the Tangent of
the, Compliment of its Diftance from the Pole
of Projection fet off in the projected Axis, on
the contrary fide of the Interfection, will give
its Centre, 2 |
_ For en is equal to the Tangent of can, the
Complement of dac. |
E 2. That
_-80 © Méfcellanea Caviofa.
2, That the Secant of the Complement of
the fame Diftance will beits Setni-diameter ;
for an equal to dz is equal to the Secant of can,
the Complement of dac....
~-g. Since wa equal. to. dn is pa ad virsanial
.. en, it follows univerfally, that the Semi-tangent
of the Diftance of any great Circle:to bé»pro-
__ jected. ( that. does not pafs through: the: Eye-
Point) from the remoter.Pole-of;Proje@ion,
_ Setoff in the projected Axison.one, fide: of the |
.. Center, will give. its Interfection 4>“and*the
..Fangent, of the Complement of the fame Di-
ftance, fet off them the fame projected:Axis on
_ the contrary. ts will giveits-Centre.
cb . In-all fmallCircles which
. Fig. 8... cut he Periphory. of :the! Plain
» . © Of ProjeGion-at« right: Angles ;
annie which is the fame thing, whofe Poles lie in
. the Gircumference.of.-the Plain of Projection,
~ that the Secant of the Complement sof their
_ .Diftances from the remoter Pole. of Projection,
_ fet of from the Genter. in‘ the- projected Axis,
__will give their Centers ; for em: is. exiak to) the
Secant of pea...
man nae §inat he Tangent, of the Garis Bilftence
,, will be the Semi-diameter 5: fors-ma ‘equal: to
md is the Tangent of pea, or the Arch” pa'the
Complement of de its Diftance from the Pole
of Projection.
6. Since cm is equal to cdand dm, it follows,
_ that the Semi-tangent of its Diftance from the
_ Pole of Projedtion, fet off in the projected Axis,
gives one Interfection; and the Fangentof the
_ Complement of the farne Diftance,fet off fron}
the Interfection the. pes ae gives the Centre, |
PRO:
: ‘a4
a ee or 5 se pordy: ye
» MISMO
} ; f a. rN ? ‘ ie & : . 4
Sha GMP. wh r SB! dK ». eX.
ee , ¢ & 2} _ e i aS § ‘4 bs
roimsih PROPOSITION HE
” ioe: ait Se sé x ; Cys 4 M4 e ‘ vt. p 4h eas. j ,
_ If through the remete Polesof = Fig. 9.
ootwo Circles, two other Circles
»obedrawn upon the Surface of the Sphere, they
yi yill cut°off equal Arches in thofe great Circles ;
»y andilikewife in the finall Circles equi-diftant
cofrom® thofe Poles through which ‘they paf.
)Letp be the Pole of the Circles md and rn, c
‘othe Poie of the Circles ab and qo; pdnob and
Ipmrqa the Reprefentation of the two Circles
© palling through the Poles p andc. I fay, ‘
1. iOf the great Circlesmd and ab, the Arches
oimmd andiab. cut off are equal. F's
je? For'the Arch pdb is equal-to chd, and pma e-
esqual to cam, and the Angle apb equal to the
| Angle bea’; wherefore the Triangle apb and mcd
ncare equal, and confequently ab equal to md.
19MR €payiiouss
tl ik Of theleffer Circles, go and rz equally di-
»/ ftant fromthe Polescand p. I fay, the Arches
| goandsrn:are equal: For the Arch caq is equal
to pmr, and cho equal to pdu, and the Angle
»geo\equalito rpn , wherefore the Triangles qco
>) andypmiare equal, and confequently the ‘Atthes
oigovand> ym are equal: q.e.d. 0°. a
an
| Whence it follows,
50) oie’ Thatif through the Pole of any projected
igreat:Circle, and any Part or Segment thereof,
oiLinescbe drawn to the Periphery of the Plain
cn@f Projection, they will cut off an Arch equal
tothe Segment of the projected Arch.
82 Mifcellanea Curiofa:
2: That if through the Pole of any final}
Circle, Lines be drawn to the Extremities of
any Segment or Portion thereof, (and con-
tinued if need be) they will cut off an Arch
equivalent to: it in that fmall Circle, which is
parallel to the Periphery of the Plain of Pro-+
jection, and as far diftant from the under Poles —
as the given Circle is diftant from its upper:
Pole:
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meyer nee ny ot
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6 ee et FP rie
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